ASPECTS OF MULTIVARIABLE OPERATOR THEORY ON WEIGHTED SYMMETRIC FOCK SPACES

H. TURGAY KAPTANOĞLU

Abstract. We obtain all Dirichlet spaces F_{q}, q ∈ ℝ, of holomorphic functions on the unit ball of ℂ^{N} as weighted symmetric Fock spaces over ℂ^{N}. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the F_{q}. We pick out the analytic Hilbert modules from among the F_{q}. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each F_{q}. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C^{∗}-algebras generated by the shift operators on the F_{q} fit in exact sequences that are in the same Ext class. We identify the groups K_{0} and K_{1} of the Toeplitz algebras on the F_{q} arising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces F_{b} that depend on a weight sequence b.

1. Introduction

The purpose of this paper is to investigate certain aspects of multivariable operator theory on a family of Hilbert spaces that are weighted symmetric Fock spaces. The spaces in this family are realized as Hilbert spaces of holomorphic functions defined on the unit ball 𝔹 of ℂ^{N} and include the standard weighted Bergman spaces, the Hardy space, the Drury-Arveson space, and the Dirichlet space as special cases. We name them the Dirichlet spaces F_{q} indexed by q ∈ ℝ. We focus on identifying the similarities and differences among various Hilbert spaces from the operator-theoretic point of view as q varies. This also highlights why and how the Drury-Arveson space is so special among the Dirichlet spaces. We concentrate mostly on N > 1, but some of our results are new even for N = 1.

Our starting point is noticing that the Drury-Arveson space A defined as a symmetric Fock space is one of the Dirichlet spaces which is the Hilbert subfamily of the Besov spaces on 𝔹. We want to investigate how much of the operator theory developed for A can be extended to the remaining Hilbert spaces and how. We do our work by obtaining all the F_{q} as weighted symmetric Fock spaces. It turns out

Date: 2 July 2013. 2010 Mathematics Subject Classification. Primary 47A13, 47B32; Secondary 19K33, 32A36,

32A37, 46E20, 46E22, 46L08, 47A20, 47A30, 47A53, 47B35, 47B37, 47B38, 47C15.

Key words and phrases. Fock, Dirichlet, Drury-Arveson, Hardy, Bergman, reproducing kernel Hilbert space, radial differential operator, multiplier, virtual point, analytic Hilbert module, shift, row contraction, von Neumann inequality, spectrum, commutant, Fredholm, hyponormal, subnormal, Toeplitz, C^{∗}-algebra, short exact sequence, extension, Busby invariant, K-groups.

that we can be even more general than the Dirichlet family by starting with a quite general weight sequence.

Our work has its beginnings in [Ar3], which deals solely with the Drury-Arveson space A which is our space F_{−N}. Several other sources deal with a family of spaces that are reproducing kernel Hilbert spaces on 𝔹 whose reproducing kernels are powers of the Bergman kernel. These correspond to q > −(1 + N) in our family of Dirichlet spaces. However, for q ≤ −(1 + N) as well, the Dirichlet spaces F_{q} are reproducing kernel Hilbert spaces on 𝔹 whose reproducing kernels are hypergeometric functions that converge on 𝔹, and this is sufficient for many applications. This subfamily has largely been ignored due to a lack of information about the reproducing kernels of the spaces although the Dirichlet space itself is F_{−(1+N)}.

Working with holomorphic function spaces on 𝔹 that are also reproducing kernel Hilbert spaces allows us to use more function theory than usual. We obtain several of our results on the Taylor (or homogeneous) series expansions of the functions in the spaces, because many interesting operators turn out to be diagonal operators. This actually simplifies our work and helps us to obtain explicit formulas. Thus many of our proofs are more concrete and thus more understandable. Rather than trying to obtain the most general results, we completely analyze the effect of q ∈ ℝ. In several places, we can easily be more general and speak of the weighted symmetric Fock spaces F_{b} depending on a weight sequence b = {b_{k}}.

Here is a synopsis of the paper. In Section 3, we obtain the generalized Dirichlet spaces as weighted symmetric Fock spaces. In Section 4, we consider those Dirichlet spaces that are Hilbert Besov spaces of holomorphic functions on 𝔹. Here radial differential operators are the central idea. In Section 5, we identify those Dirichlet spaces that are analytic Hilbert modules. In Section 6, we obtain many formulas for the shift operators acting on the Dirichlet spaces that we need later. We notice that they are very similar to radial differential operators. In Section 7, we prove that there is a von Neumann inequality for row contractions on any Hilbert space with respect to any Dirichlet space using a dilation theorem. In Section 8, we investigate the spetra of shift operators in some detail when N = 1, because then they turn out to be weighted shifts. It is here that we also determine the commutants of the shift operators in all dimensions. In Section 9, we investigate how much of various kinds of almost normality is present in the shift operators on the Dirichlet spaces. In Section 10, we show that the C^{∗}-algebra generated by the shift operators on each Dirichlet space fits in a short exact sequence. This in turn yields information about the Fredholm properties of the elements of the C^{∗}-algebra and other related objects. In the final Section 11, we show that all the short exact sequences obtained are in the same Ext class as well as computing the K-theory groups K_{0} and K_{1} of these C^{∗}-algebras.

Acknowledgements. The author dedicates this paper to the memory of Walter Rudin and of Mary Ellen Rudin who wrote their only joint paper [RR] after their retirement from teaching.

The author thanks Baruch Solel of the Technion for asking the right question, Aurelian Gheondea of Bilkent University for several useful discussions, Joseph A. Ball of Virginia Tech for updating him on the literature, Thomas L. Kriete and Barbara D. MacCluer of the University of Virginia for their wonderful lectures, and an anonymous referee for pointing out a significant gap in a proof.

The author is deeply indebted to and generously thanks Semra Öztürk Kaptanoğlu of Middle East Technical University for the creative discussions that made the mathematical content of this paper considerably richer.

2. Notation

We start by introducing the basic notation. The usual Hermitian inner product in ℂ^{N} is ⟨z, w⟩ = z_{1}\overline{w}_{1} + ⋯ + z_{N} \overline{w}_{N} with the associated norm |z| = \sqrt{⟨z, z⟩}, where the overline represents the complex conjugate. This gives rise to the orthonormal basis {e_{1}, …, e_{N}} of ℂ^{N}, where e_{j} = (0, …, 0, 1, 0, …, 0) with 1 in the jth position. We use multi-index notation in which α = (α_{1}, …, α_{N}) ∈ ℕ^{N} is an N-tuple of nonnegative integers, |α| = α_{1} + ⋯ + α_{N}, α! = α_{1}! ⋯ α_{N}!, z^{α} = z_{1}^{α_{1}} ⋯ z_{N}^{α_{N}}, and 0^{0} = 1, where z = (z_{1}, …, z_{N}) ∈ ℂ^{N}.

We denote the unit ball of ℂ^{N} with respect to the usual norm by 𝔹 and its boundary the unit sphere by ∂𝔹. When N = 1, the unit ball is the unit disc 𝔻. We denote the space of holomorphic functions on 𝔹 by H(𝔹) and the algebra of bounded holomorphic functions on 𝔹 by H^{∞}. The ball algebra A(𝔹) consists of holomorphic functions on 𝔹 that extend continuously to its closure \overline{𝔹}. We denote the space of continuous functions on a compact set K by C(K). The last three spaces are adorned with the supremum norm.

Let ν be the volume measure on ℂ^{N} normalized as ν(𝔹) = 1. For q ∈ ℝ, we define on 𝔹 also the measures

dν_{q}(z) = C_{q} (1 − |z|^{2})^{q} dν(z).

These measures are finite only for q > −1 and then we choose C_{q} so that ν_{q}(𝔹) = 1. For q ≤ −1, we set C_{q} = 1. For 0 < p < ∞, we denote the Lebesgue classes with respect to ν_{q} by L_{q}^{p} and the sequence spaces whose pth powers are summable by ℓ^{p}.

The Pochhammer symbol (x)_{y} for x, y ∈ ℂ is defined by

(x)_{y} = \frac{Γ(x + y)}{Γ(x)}

when x and x+y are off the pole set −ℕ of the gamma function Γ. This is a shifted factorial since (x)_{k} = x(x + 1) ⋯ (x + k − 1) for positive integer k. In particular, (1)_{k} = k! and (x)_{0} = 1. Stirling formula gives

(1) \frac{Γ(c + x)}{Γ(c + y)} ∼ c^{x−y},  \frac{(x)_{c}}{(y)_{c}} ∼ c^{x−y},  and  \frac{(c)_{x}}{(c)_{y}} ∼ c^{x−y}  (Re c → ∞),

where A ∼ B means that |A/B| is bounded above and below by two positive constants. Such constants that are independent of the parameters in the equation are all denoted by the generic unadorned upper case C.

The classical hypergeometric function is

2F_{1}(a, b; c; z) = ∑_{k=0}^{∞} \frac{(a)_{k} (b)_{k}}{(c)_{k} (1)_{k}} z^{k} (a, b ∈ ℝ, c > 0, z ∈ 𝔻).

The series converges absolutely and uniformly for z in compact subsets of 𝔹; therefore _{2}F_{1} ∈ H(𝔹).

Consider the space P_{k} of all holomorphic polynomials that are homogeneous of degree k. Elements of P_{k} have the form ∑_{|α|=k} c_{α}z^{α}. The dimension of P_{k} is

(2)  δ_{k} = \frac{(N)_{k}}{k!} ∼ k^{N−1}  (k → ∞).

We let P_{k} denote the orthogonal projection from a larger Hilbert space onto P_{k}.

The algebra of all bounded operators on a complex separable Hilbert space H is denoted B(H) and of all compact operators K(H). The spectrum and the point spectrum (the set of eigenvalues) of an operator T ∈ B(H) are denoted σ(T) and σ_{p}(T), respectively, as usual. If the inner product of H is [·, ·], T is called positive and we write T ≥ 0 if [T v, v] ≥ 0 for all v ∈ H. For a, b ∈ H, a ⊗ b denotes the rank-1 operator defined by (a ⊗ b)(v) = [v, b] a for v ∈ H.

The commutant {⋯ }′ of a set {⋯ } of bounded operators on H is the set of all bounded operators on H that commute with each operator in {⋯ }, and is an algebra containing the identity operator I. The double commutant {⋯ }′′ is the commutant of the commutant. A net of bounded operators {T_{ℓ}} on H converges to T ∈ B(H) in the strong operator topology (sot) if T_{ℓ}v → T v for all v ∈ H; the convergence is in the weak operator topology (wot) if [T_{ℓ}v, u] → [T v, u] for all v, u ∈ H.

The singular values of T are the eigenvalues of the positive operator (T^{∗}T)^{1/2}, where T^{∗} is the adjoint of T. For 0 < p < ∞, T belongs to the Schatten class C^{p}(H) if its sequence of singular values belongs to ℓ^{p}. Of course, C^{p}(H) ⊂ K(H), and K(H) and C^{p}(H) are ideals of B(H). The Banach space B(H) can be realized as the dual of C^{1}(H), and the topology on B(H) induced by this duality is called the weak^{∗} (ultraweak) topology of B(H). The quotient Q(H) := B(H)/K(H) is called the Calkin algebra. The essential spectrum σ_{e}(T) and the essential norm ∥T ∥_{e} of T are the spectrum and the norm of the coset [ T ] := T + K(H) of T in Q(H), respectively.

As usual, the abbreviations dim, ker, and im denote the dimension, kernel, and image (range). We use terms like positive and increasing loosely to mean nonnegative and nondecreasing. The right hand side of := defines its left hand side.

3. Weighted Symmetric Fock Spaces

The symmetric tensor product allows one to carry out multivariable operator theory by pretending working with a single variable. We modify and generalize this construction by adding appropriate weights.

Following the notation of [Ar3], let E = ℂ^{N} and denote by E^{k} the subspace of the tensor product E^{⊗k} of k copies of E which is fixed elementwise under the action of the permutation group on k objects. So E^{k} is spanned by the set { z^{⊗k} : z ∈ E }. For completeness we set E^{0} = ℂ and z^{⊗0} = 1. The usual inner product and norm on E^{k} are the restrictions of those on E^{⊗k} and are given on elementary tensors by ⟨z^{⊗k}, w^{⊗k}⟩_{k} = ⟨z, w⟩^{k} and |z^{⊗k}|_{k} = |z|^{k}, but we need to introduce weights to accomodate a whole family of spaces.

For this, we start with a weight sequence b = { b_{k} : k = 0, 1, 2, … } of strictly positive real numbers with b_{0} = 1 and satisfying

(3)  Z_{b} := \frac{1}{lim sup_{k→∞} b_{k}^{1/k}} ≥ 1.

Note that (3) implies nothing about the boundedness, summability, or monotoneness of b. The coefficient sequence of any power series in one variable whose disc of convergence includes 𝔻 is acceptable for b. This kind of b is more general than what is called regular in [GHX, p. 231] by [R, Example 3.35]. In most practical cases, b_{k} acts like a fixed power of k.

We endow E^{k} with the weighted inner product defined on elementary tensors by

[z^{⊗k}, w^{⊗k}]_{k} := b_{k} ⟨z^{⊗k}, w^{⊗k}⟩_{k} = b_{k} ⟨z, w⟩^{k}

and extended linearly to all of E^{k}. Thus E^{⊗k} and E^{k} are Hilbert spaces of dimensions N^{k} and N, respectively. Let’s denote the corresponding norm on E^{k} by ∥·∥_{k}; so

∥z^{⊗k}∥_{k}^{2} := b_{k} ⟨z^{⊗k}, z^{⊗k}⟩_{k} = b_{k} |z|^{2k}.

Every holomorphic homogeneous polynomial f_{k} : ℂ^{N} → ℂ of degree k gives rise to a unique linear functional f̃_{k} : E^{k} → ℂ by f̃_{k}(z^{⊗k}) = f_{k}(z). There exists a unique ζ_{k} ∈ E^{k} such that f̃_{k}(z^{⊗k}) = [z^{⊗k}, ζ_{k}]_{k} for all z ∈ ℂ^{N} by the Riesz representation theorem. In particular, ζ_{0} = \overline{f}_{0}. For the monomial f_{k}(z) = z_{1}^{k}, we compute easily that ζ_{k} = (b_{k}^{−1/k}, 0, …, 0)^{⊗k}. If f is a polynomial of degree n with homogeneous expansion f = f_{0} + f_{1} + ⋯ + f_{n}, then

\begin{aligned}f(z) & = [1, ζ_{0}]_{0} + [z, ζ_{1}]_{1} + [z ⊗ z, ζ_{2}]_{2} + ⋯ + [z^{⊗n}, ζ_{n}]_{n} \\ & = \overline{ζ}_{0} + b_{1} ⟨z, ζ_{1}⟩ + b_{2} ⟨z ⊗ z, ζ_{2}⟩_{2} + ⋯ + b_{n} ⟨z^{⊗n}, ζ_{n}⟩_{n} (z ∈ ℂ^{N}). \end{aligned} ; (4)

This representation induces on polynomials an inner product [·, ·]_{b} whose associated norm ∥ · ∥_{b} is given by

∥f∥_{b}^{2} := ∥ζ_{0}∥_{0}^{2} + ∥ζ_{1}∥_{1}^{2} + ⋯ + ∥ζ_{n}∥_{n}^{2} = |ζ_{0}|^{2} + b_{1} |ζ_{1}|_{1}^{2} + ⋯ + b_{n} |ζ_{n}|_{n}^{2}.

By construction, ∥ζ_{0}∥ = |f_{0}| and [f_{k}, f_{l}]_{b} = 0 if k ≠ l.

Definition 3.1. We denote by F_{b} the Hilbert space obtained by completing the polynomials in the norm ∥ · ∥_{b}.

So ℂ ⊂ F_{b} and F_{b} is separable. Next we identify F_{b} with the weighted symmetric Fock space

E_{b}(E) = E^{0} ⊕ E^{1} ⊕ E^{2} ⊕ ⋯

(called also a bosonic Fock space by physicists) over E, where each E^{k} is taken with the norm ∥ · ∥_{k}. A function f ∈ F_{b} has the Taylor (homogeneous) expansion

(5)  f(z) = ∑_{k=0}^{∞} [z^{⊗k}, ζ_{k}]_{k} = ∑_{k=0}^{∞} b_{k} ⟨z^{⊗k}, ζ_{k}⟩_{k} = ∑_{k=0}^{∞} f_{k}(z),

where Gf := (ζ_{0}, ζ_{1}, ζ_{2} …) ∈ E_{b}(E) is the sequence of Taylor coefficients. Starting with (ζ_{0}, ζ_{1}, ζ_{2} …), an f it defines belongs to F_{b} if and only if

(6)  ∥f∥_{b}^{2} := ∑_{k=0}^{∞} ∥f_{k}∥_{k}^{2} = ∑_{k=0}^{∞} ∥ζ_{k}∥_{k}^{2} = ∑_{k=0}^{∞} b_{k} |ζ_{k}|_{k}^{2} < ∞.

The right hand side of this equation also defines the corresponding norm ∥ · ∥_{b} on E_{b}(E). Because the Taylor coefficients of f appear in the second position in each term in (5), the correspondence f ↦ Gf is conjugate linear and then we have [f, g]_{b} = [Gg, Gf]_{b}. With this identification, any homogeneous polynomial f_{k} of degree k can be considered as an element of E^{k} since Gf_{k} has only one nonzero term which is in E^{k}. Hence f_{k}(z) = z^{α} ∈ E^{k} if |α| = k.

Further, we define

(7)  K_{b}(z, w) := ∑_{k=0}^{∞} b_{k} ⟨z, w⟩^{k},

where the series converges absolutely for z, w ∈ 𝔹 and uniformly on compact subsets by (3). Then

|f(z)| ≤ ∑_{k=0}^{∞} b_{k} |⟨z^{⊗k}, ζ_{k}⟩_{k}| ≤ ∑_{k=0}^{∞} b_{k} |z^{⊗k}|_{k} |ζ_{k}|_{k} = ∑_{k=0}^{∞} \sqrt{b_{k}} |z|^{k} \sqrt{b_{k}} |ζ_{k}|_{k}

≤ (∑_{k=0}^{∞} b_{k} |z|^{2k})^{1/2}(∑_{k=0}^{∞} b_{k} |ζ_{k}|_{k}^{2})^{1/2} = \sqrt{K_{b}(z, z)} ∥f∥_{b}.

Observe that

(8)  K_{b}(z, z) ≥ 1  (z ∈ 𝔹)

since we have picked b_{k} > 0 for all k and b_{0} = 1. Let’s sum up.

Theorem 3.2. An element f of the space F_{b} is a holomorphic function on 𝔹. It has the Taylor expansion (5) on 𝔹 which converges absolutely and uniformly on compact subsets of 𝔹. The evaluation of f at z ∈ 𝔹 is a bounded linear functional on F_{b} of norm at most \sqrt{K_{b}(z, z)}.

Noting that

K_{b}(z, w) = ∑_{k=0}^{∞} b_{k} ⟨z, w⟩^{k} = ∑_{k=0}^{∞} b_{k} ⟨z^{⊗k}, w^{⊗k}⟩_{k} = ∑_{k=0}^{∞} [z^{⊗k}, w^{⊗k}]_{k}

and comparing with (4), we see that GK_{b}(·, w) = (1, w, w ⊗ w, w^{⊗3}, …); that is, ζ_{k} = w^{⊗k} for k = 0, 1, 2, … for K_{b}(·, w). Since

∥GK_{b}∥_{b}^{2} = ∑_{k=0}^{∞} ∥w^{⊗k}∥_{k}^{2} = ∑_{k=0}^{∞} b_{k} |w|^{2k} < ∞ (w ∈ 𝔹)

by (3), we conclude that K_{b}(·, w) ∈ F_{b} for every w ∈ 𝔹. Moreover,

(9)  [f(·), K_{b}(·, w)]_{b} = ∑_{k=0}^{∞} [w^{⊗k}, ζ_{k}]_{k} = f(w)  (f ∈ F_{b}, w ∈ 𝔹)

by (5). This shows that if f ∈ F_{b} and [f(·), K_{b}(·, w)]_{b} = 0 for all w ∈ 𝔹, then f = 0; that is, { K_{b}(·, w) : w ∈ 𝔹 } is dense in F_{b}. Also

[K_{b}(·, z), K_{b}(·, w)]_{b} = ∑_{k=0}^{∞} [w^{⊗k}, z^{⊗k}]_{k} = ∑_{k=0}^{∞} b_{k} ⟨w, z⟩^{k} = K_{b}(w, z) (z, w ∈ 𝔹).

In particular,

(10)  ∥K_{b}(·, w)∥_{b} = \sqrt{K_{b}(w, w)}.

We have proved the following.

Theorem 3.3. The space F_{b} is a reproducing kernel Hilbert space with reproducing kernel K_{b}(z, w).

When {b_{k}} ∈ ℓ^{1}, that is, when

(11)  ∑_{k=0}^{∞} b_{k} < ∞,

K_{b}(z, w) converges uniformly and absolutely for (z, w) ∈ \overline{𝔹}×\overline{𝔹}. By (9), this implies that an f ∈ F_{b} extends continuously to \overline{𝔹}, and thus F_{b} ⊂ A(𝔹). If (11) does not hold for a {b_{k}}, then K_{b} is unbounded and consequently F_{b} contains unbounded functions.

The kth Taylor coefficient ζ_{k} of a given f ∈ F_{b} is a finite sum (over m) of terms of the form w_{m}^{⊗k} with w_{m} ∈ ℂ^{N}. Then

\begin{aligned}f(z) & = ∑_{k=0}^{∞} [z^{⊗k}, ζ_{k}]_{k} = ∑_{k=0}^{∞} b_{k} ∑_{m} ⟨z, w_{m}⟩^{k} = ∑_{k=0}^{∞} b_{k} ∑_{m} ∑_{|α|=k} \frac{k!}{α!} z^{α} \overline{w}_{m}^{α} \\ & = ∑_{k=0}^{∞} ∑_{|α|=k}(b_{k} \frac{k!}{α!} ∑_{m} \overline{w}_{m}^{α})z^{α} = ∑_{k=0}^{∞} ∑_{|α|=k} f_{α} z^{α} (z ∈ 𝔹), \end{aligned} ; (12)

where the parentheses give the coefficient f_{α} of z^{α} in terms of the w_{m}. Consider now f = K_{b}(·, w) for which ζ_{k} = w^{⊗k}, a single term, for all k. Then

∥f∥_{b}^{2} = ∑_{k=0}^{∞} ∥ζ_{k}∥_{k}^{2} = ∑_{k=0}^{∞} b_{k} ⟨w^{⊗k}, w^{⊗k}⟩_{k} = ∑_{k=0}^{∞} b_{k} ⟨w, w⟩^{k} = ∑_{k=0}^{∞} ∑_{|α|=k} b_{k} \frac{k!}{α!} |w^{α}|^{2}

= ∑_{k=0}^{∞} ∑_{|α|=k} b_{k}^{2} \frac{(k!)^{2}}{(α!)^{2}} |w^{α}|^{2} \frac{α!}{b_{k} k!} = ∑_{k=0}^{∞} ∑_{|α|=k} |f_{α}|^{2} ∥z^{α}∥_{b}^{2}.

Thus

(13)  ∥z^{α}∥_{b}^{2} = ∥z^{α}∥_{k}^{2} = \frac{α!}{b_{|α|} |α|!}.

Consequently, an f(z) = ∑_{α} f_{α} z^{α} ∈ H(𝔹) belongs to F_{b} if and only if

(14)  ∑_{α} \frac{|f_{α}|^{2}}{b_{|α|}} \frac{α!}{|α|!} < ∞.

The above computation as a byproduct also gives the orthogonality

(15)  [z^{α}, z^{β}]_{b} = 0  (α ≠ β),

even when |α| = |β| or |α| = 0. Polarizing the formula for ∥f∥_{b}, we also obtain an explicit expression for the inner product in F_{b} in terms of the Taylor coefficients:

[f, g]_{b} = ∑_{k=0}^{∞} ∑_{|α|=k} f_{α} \overline{g}_{α} ∥z^{α}∥_{b}^{2} = ∑_{k=0}^{∞} \frac{1}{b_{k} k!} ∑_{|α|=k} α! f_{α} \overline{g}_{α}.

Proposition 3.4.

\{ O_{α}^{b}(z) = \sqrt{\frac{b_{|α|} |α|!}{α!}} z^{α} : α ∈ ℕ^{N} \}

is an orthonormal basis for F_{b}.

Many of the operators we encounter are a class of diagonal operators acting on Dirichlet spaces. We collect their main properties in the next theorem and use them often without mention.

Theorem 3.5. Let {a_{k}} be a sequence of complex numbers and D : F_{b} → F_{b} be

an operator that acts on f ∈ F_{b} given by (5) by way of Df = ∑_{k=0}^{∞} a_{k}f_{k}. Then the

following hold.

(i) D is a diagonal operator and D = ∑_{k=0}^{∞} a_{k}P_{k}.

(ii) The eigenvalues of D are the a_{k} with eigenspaces P_{k} of dimension δ_{k}, and σ(D) = \overline{\{ a_{k} : k ∈ ℕ \}}.

(iii) D is invertible if and only if every a_{k} ≠ 0, and then D^{−1}f = ∑_{k=0}^{∞} a_{k}^{−1}f_{k}.

(iv) D is bounded if and only if {a_{k}} is bounded, and then ∥D∥ = sup_{k} {a_{k}}.

(v) D is compact if and only if a_{k} → 0 as k → ∞.

(vi) The adjoint of D is given by D^{∗}f = ∑_{k=0}^{∞} \overline{a}_{k} f_{k}, so D is self-adjoint if and

only if every a_{k} ∈ ℝ.

(vii) D is positive if and only if every a_{k} ≥ 0 with D^{1/2}f = ∑_{k=0}^{∞} \sqrt{a_{k}} f_{k}.

(viii) The singular values of a compact D are {|a_{k}|} each of which is repeated δ_{k} times, and for 0 < p < ∞, D belongs to C^{p}(F_{b}) if and only if {a_{k} δ_{k}^{1/p}} ∈ ℓ^{p} if and only if {a_{k} k^{(N−1)/p}} ∈ ℓ^{p}.

Proof. Consider the following order relation ≺ on the set of all multi-indices. If |α| < |β|, let α ≺ β; and among all multi-indices α with the same |α|, let ≺ be any order, say, the lexicographic order. Note that f_{k} consists of all terms containing z^{α} with the same |α| = k and all these terms are matched with the same a_{k} in D. If

f ∈ F_{b}, then f(z) = ∑_{α} f_{α} z^{α} = ∑_{α} f̃_{α} O_{α}^{b}(z) and

(16)  Df = ∑_{α} a_{|α|} f̃_{α} O_{α}^{b}.

Now almost all the conclusions of the theorem are clear by Proposition 3.4; only for (viii), we note that

∑_{α} |a_{|α|}|^{p} = ∑_{k=0}^{∞} |a_{k}|^{p} ∑_{|α|=k} 1 = ∑_{k=0}^{∞} |a_{k}|^{p} δ_{k} ∼ ∑_{k=0}^{∞} |a_{k}|^{p} k^{N−1} = ∑_{k=0}^{∞}(|a_{k}| k^{(N−1)/p})^{p}

using (2). □

4. Dirichlet Spaces

Our interest in weighted symmetric Fock spaces F_{b} arises from the well-known spaces corresponding to the special cases of the weight sequence {b_{k}}.

Definition 4.1. For q ∈ ℝ, the Dirichlet space F_{q} is the reproducing kernel Hilbert space with reproducing kernel K_{q}(z, w) given by (7) when the weight sequence is

b_{k}(q) := \begin{cases}\frac{(1 + N + q)_{k}}{k!} , & if q > −(1 + N), \\ \frac{k!}{(1 − (N + q))_{k}} , & if q ≤ −(1 + N). \end{cases}

We use the subscript q instead of b when referring to these spaces, and q ∈ ℝ is unrestricted. The first thing to note is that

(17)  b_{k}(q) ∼ k^{N+q}  (k → ∞, q ∈ ℝ)

by (1) so that we have lim_{k→∞} b_{k}(q)^{1/k} = 1 =: 1/Z_{q} and thus (3) is satisfied. Also for

all q ∈ ℝ, b_{0}(q) = 1 = ∥1∥_{q}, and for |α| > 0,

\begin{aligned}∥z^{α}∥_{q}^{2} & = \begin{cases}\frac{α!}{(1 + N + q)_{|α|}} , & if q > −(1 + N), \\ \frac{(1 − (N + q))_{|α|} α!}{(|α|!)^{2}}, & if q ≤ −(1 + N), \end{cases} \\ & ∼ \frac{1}{|α|^{N+q}} \frac{α!}{|α|!} (q ∈ ℝ). \end{aligned} ; (18)

Hence (14) takes the form

∑_{|α|>0} \frac{|f_{α}|^{2}}{|α|^{N+q}} \frac{α!}{|α|!} < ∞

as the defining condition for f(z) = ∑_{α} f_{α} z^{α} ∈ H(𝔹) ∩ F_{q}. By Definition 3.1, every

F_{q} contains the polynomials densely. If q < −(1 + N), then {b_{k}(q)} ∈ ℓ^{1} by (17), and by the discussion following (11), F_{q} ⊂ A(𝔹).

The reason for considering the particular b_{k}(q) is that special values of q give many well-known spaces. Note that

K_{q}(z, w) = \begin{cases}\frac{1}{(1 − ⟨z, w⟩)^{1+N+q}} = _{2}F_{1}(1 + N + q, 1; 1; ⟨z, w⟩), & if q > −(1 + N), \\ _{2}F_{1}(1, 1; 1 − (N + q); ⟨z, w⟩), & if q ≤ −(1 + N). \end{cases}

In particular,

K_{−(1+N)}(z, w) = \frac{1}{⟨z, w⟩} log \frac{1}{1 − ⟨z, w⟩}.

Thus

F_{q} = \begin{cases}A_{q}^{2} (weighted Bergman spaces), & if q > −1, \\ H^{2} (Hardy space), & if q = −1, \\ A (Drury-Arveson space), & if q = −N, \\ D (Dirichlet space), & if q = −(1 + N). \end{cases}

We use the notation ∥ · ∥_{A} and ∥ · ∥_{−N} interchangably, and similarly for the other spaces with special names.

Corollary 4.2. Each F_{q}, and in particular each of A_{q}^{2}, H^{2}, A, and D is a weighted symmetric Fock space over ℂ^{N}.

Proof. Theorems 3.2 and 3.3 apply. □

Note that the case of the Drury-Arveson space (q = −N) is the simplest, because it is unweighted due to b_{k}(−N) = 1 for all k.

Remark 4.3. There are two immediate critical values of q from the point of view of the b_{k}(q). It is easy to check that {b_{k}(q)} is an increasing sequence for q ≥ −N, and it is a decreasing sequence for q < −N. By contrast, {b_{k}(q)} satisfies (11) and F_{q} ⊂ A(𝔹) if and only if q < −(1 + N). But q = −1 is also critical. By [BB, Theorem 4.1], there are functions in A(𝔹) that are not in F_{b} if and only if q < −1. We have many more results of this kind.

To obtain one more description of the spaces F_{q}, we recall that the radial deriv-

ative of an f ∈ H(𝔹) given by its homogeneous expansion (5) is Rf = ∑_{k=1}^{∞} kf_{k}. The

radial derivative is also called the number operator by physicists since it represents the number of particles in a physical system. For all s, t ∈ ℝ, we define additional radial fractional differential operators D_{s}^{t} of order t ∈ ℝ on H(𝔹) by

(19)  D_{s}^{t}f := ∑_{k=0}^{∞} d_{k}(s, t) f_{k} := ∑_{k=0}^{∞} \frac{b_{k}(s + t)}{b_{k}(s)} f_{k}.

These operators are defined by making use of the extra flexibility provided by the parameter q in b_{k}(q) which does not exist in general F_{b}. The coefficients d_{k}(s, t) have uniform growth rate in α with |α| fixed. This particular way of defining the d_{k}(s, t) and thus the D_{s}^{t} in terms of the weight sequence {b_{k}} first appears in [K2, Definition 7.1]. Then

D_{q}^{t}K_{q}(z, w) = K_{q+t}(z, w) (q, t ∈ ℝ),

where the D_{q}^{t} act on the holomorphic variable z.

The D_{s}^{t} clearly commute with each other and with R. The essential properties of D_{s}^{t} are that

d_{k}(s, t) > 0  (k = 0, 1, 2, …)  and  d_{k}(s, t) ∼ k^{t}  (k → ∞).

Other properties are that

(20) D_{s}^{0} = I,  D_{s+t}^{u}D_{s}^{t} = D_{s}^{u+t},  and  (D_{s}^{t})^{−1} = D_{s+t}^{−t}  (s, t, u ∈ ℝ).

In particular, D_{s}^{t}(1) = 1 and D_{−N}^{1} = I + R. For comparison, R is not invertible. Many of these claims as well as our next result follow directly from Theorem 3.5.

Proposition 4.4. Every D_{s}^{t} : F_{b} → F_{b} (and R) is a positive operator. It is bounded if and only if t ≤ 0 and compact if and only if t < 0. The same equivalent conditions applied on the total order characterize the boundedness and compactness of also a composition of several of the D_{s}^{t}. Moreover, D_{s}^{t} ∈ C^{p}(F_{q}) if and only if N + pt < 0. So (I + R)^{−1} = D_{1−N}^{−1} ∈ C^{p}(F_{q}) if and only if p > N.

Proof. Only the Schatten class membership requires some comment. Now the singular values are |a_{k}| = d_{k}(s, t) ∼ k^{t} and the condition in Theorem 3.5 (viii) is that the sequence {(k^{t})^{p} k^{N−1}} be summable. □

It is interesting that the conditions in Proposition 4.4 do not depend on q. The result about the Schatten class membership of (I + R)^{−1} acting only on A is given in [Ar3, Appendix A] with a more complicated proof.

Corollary 4.5. The map D_{s}^{t} : F_{q_{1}} → F_{q_{2}} is bounded if and only if t ≤ (q_{2} − q_{1})/2 and compact if and only if t < (q_{2} − q_{1})/2. It belongs to C^{p}(F_{q_{1}}→F_{q_{2}}) if and only if p > 2N/(q_{2} − q_{1} − 2t). In particular, the inclusion map i : F_{q_{1}} → F_{q_{2}} is bounded if and only if q_{1} ≤ q_{2}, compact if and only if q_{1} < q_{2}, and belongs to C^{p}(F_{q_{1}}→F_{q_{2}}) if and only if p > 2N/(q_{2} − q_{1}).

Proof. By Proposition 3.4 and (17), O_{α}^{q_{1}} ∼ k^{(q_{1}−q_{2})/2} O_{α}^{q_{2}}. Then

D_{s}^{t}f ∼ ∑_{α} |α|^{t} f̃_{α} O_{α}^{q_{1}} ∼ ∑_{α} |α|^{t} |α|^{(q_{1}−q_{2})/2} f̃_{α} O_{α}^{q_{1}} ∼ ∑_{k=0}^{∞} k^{t+(q_{1}−q_{2})/2} f_{k}.

This makes clear the assertions about D_{s}^{t}. For i, we just set t = 0. □

To form stronger connections with function theoretical techniques, we bring out another characterization of the F_{q} as Sobolev-type spaces.

Theorem 4.6. Define I_{s}^{t}f(z) := (1 − |z|^{2})^{t} D_{s}^{t}f(z). An f ∈ H(𝔹) belongs to F_{q} if and only if I_{s}^{t}f belongs to L_{q}^{2} for some s, t satisfying

(21)  q + 2t > −1.

That is, I_{s}^{t} : F_{q} → L_{q}^{2} is an imbedding for any s, t satisfying (21). The L_{q}^{2} norm of any such I_{s}^{t}f and the weighted symmetric Fock space norm ∥f∥_{q} are equivalent.

Proof. See [K1, Corollary 4.2, Theorem 4.3]. □

Thus for any q ∈ ℝ, F_{q} is the Besov space B_{q}^{2}, where the parameters of B_{q}^{2} follow the pattern of [K1, p. 386].

Remark 4.7. The equivalent condition stated in Theorem 4.6 for the membership of an f ∈ H(𝔹) in F_{q} is

(22)  ⫴f⫴_{q}^{2} := C_{q} ∫_{𝔹} |D_{s}^{t}f(z)|^{2} (1 − |z|^{2})^{q+2t} dν(z) < ∞

for any s, t satisfying (21). Note that we have a whole family of equivalent norms for each such s, t. Polarization gives again equivalent inner products on the F_{q}:

⟦f, g⟧_{q} := C_{q} ∫_{𝔹} D_{s}^{t}f(z) \overline{D_{s}^{t}g(z)} (1 − |z|^{2})^{q+2t} dν(z).

If q > −1, then by (21), we can choose t = 0, no differentiation on f is required in (22), and in such a case ⫴f⫴_{q} = ∥f∥_{L_{q}^{2}}. Equivalently, for q > −1, I_{s}^{t} can be taken as the inclusion map and hence F_{q} ⊂ L_{q}^{2}.

The question of which F_{q} can be characterized by an integral norm of f itself is answered negatively for q = −N in [Ar3, p. 180]. This is extended negatively to all q < −1 in [BB, Theorem 4.3] by considering those F_{q} as Hardy-Sobolev spaces. The answer is clearly positive also for q = −1 using the surface measure on ∂𝔹; so in this case F_{−1} ⊂ L^{2}(∂𝔹).

Membership in F_{q} can still be decided by an equivalent norm given by an integral of f itself, but only under a limit; see [AK1, Theorem 9] for q = −N and [AK2, Proposition 2.1] for q ≥ −(1 + N).

Corollary 4.8. For any q, s, t, D_{s}^{t}(F_{q}) = F_{q+2t}, and isometrically with appropriate values of the parameters.

Proof. For f ∈ F_{q}, let g = D_{s}^{t}f ∈ F_{q+2t}, and choose u so that q + 2(t + u) > −1. Then the norm ⫴f⫴_{q} using D_{s}^{t+u} and the norm ⫴g⫴_{q} using D_{s+t}^{u} in (22) are equal. Thus D_{s}^{t} : F_{q} → F_{q+2t} is an isometry with the chosen values of the parameters. Throughout we have used (20). □

5. Multipliers and Analytic Hilbert Modules

A function g ∈ H(𝔹) is called a multiplier for a Hilbert space H of holomorphic functions on 𝔹 if gf ∈ H whenever f ∈ H. The collection M(H) of all multipliers of H is a Banach algebra containing I. The natural norm ∥ · ∥_{M(H)} on M(H) is the norm of the operator M_{g} : H → H of multiplication by g.

We have M(F_{b}) ⊂ F_{b} for all b since 1 ∈ F_{b}. Also, since ∥1∥_{b} = 1, if g ∈ M(F_{b}), then ∥g∥_{b} = ∥g1∥_{b} = ∥M_{g}1∥ ≤ ∥M_{g}∥ ∥1∥_{b} = ∥M_{g}∥ = ∥g∥_{M(F_{b})}. Further, by [BB, Lemmas 5.1 and 5.2] (and by [Ar3, Proposition 2.2] for the case F_{−N}), M(F_{b}) ⊂ H^{∞} and ∥g∥_{H^{∞}} ≤ ∥g∥_{M(F_{b})} for g ∈ M(F_{b}). The reverse inequalities in the last two sentences are not true for all b however. In any case, M(F_{b}) ⊂ F_{b}∩H^{∞}.

Moreover, the well-known quick computation

[f(·), M_{g}^{∗}K_{b}(·, w)]_{b} = [(gf)(·), K_{b}(·, w)]_{b} = g(w)f(w) = [f(·), \overline{g(w)}K_{b}(·, w)]_{b}

for any w ∈ 𝔹 and f ∈ F_{b} shows that

(23)  M_{g}^{∗}K_{b}(·, w) = \overline{g(w)}K_{b}(·, w),

that is, \overline{g(w)} for w ∈ 𝔹 is an eigenvalue of M_{g}^{∗} with eigenvector K_{b}(·, w).

It is clear by the t = 0 case of Remark 4.7 that M(F_{q}) = H^{∞} for q ≥ −1 with equal norms. But for other values of q, the inclusion of the multiplier algebra in H^{∞} is proper. By [BB, Theorem 4.2], for q < −1, there are functions in A(𝔹) that are not in M(F_{q}).

If q < −(1 + N), then F_{q} an algebra and M(F_{q}) = F_{q}; see [OF, Theorem 3.8]. For −(1 + N) ≤ q < −1, characterizations of M(F_{q}) are not so simple. See, for example, [OF, Theorems 3.7 and 3.10].

The spaces F_{b} include all holomorphic polynomials in N variables densely, and these polynomials are multipliers for all F_{b}. Thus each F_{b} is a module over the polynomial algebra ℂ[z] = ℂ[z_{1}, …, z_{N}].

The following definitions are taken from [CG, Section 2.2].

Definition 5.1. A point w ∈ ℂ^{N} is called a virtual point of a reproducing kernel Hilbert space H of holomorphic functions on 𝔹 if the point evaluation functional at w ∈ 𝔹 extends boundedly from ℂ[z] to all of H. The space H is called an analytic Hilbert module if 1 ∈ H, polynomials are dense in H, the multiplier algebra M(H) of H contains ℂ[z], and the set of virtual points of H coincides with 𝔹.

Theorem 5.2. The space F_{b} is an analytic Hilbert module if and only if {b_{k}} ∉ ℓ^{1}.

Proof. We only have to check the virtual points. We know that points of 𝔹 are virtual points by Theorem 3.2.

First suppose {b_{k}} ∉ ℓ^{1}, and that z_{0} ∈ ℂ^{N} with |z_{0}| ≥ 1 is a virtual point of F_{b}. Then there is a C depending on z_{0} such that |K_{b}(z_{0}, w)| ≤ C ∥K_{b}(·, w)∥_{b} for all small w, that is,

| ∑_{k=0}^{∞} b_{k} ⟨z_{0}, w⟩^{k}|^{2} ≤ C ∑_{k=0}^{∞} b_{k} |w|^{2k} (|w| < 1/|z_{0}|)

by (10). Let w = tz_{0} with t ≥ 0 so small that t^{2} < t; then

( ∑_{k=0}^{∞} b_{k} t^{k} |z_{0}|^{2k})^{2} ≤ C ∑_{k=0}^{∞} b_{k} t^{2k} |z_{0}|^{2k} ≤ C ∑_{k=0}^{∞} b_{k} t^{k} |z_{0}|^{2k} (0 ≤ t < \frac{1}{|z_{0}|^{2}} ).

After an obvious cancellation and letting t → 1/|z_{0}|^{2} from the left, we obtain

∑_{k=0}^{∞} b_{k} ≤ C

contrary to assumption. Thus z_{0} cannot be a virtual point of F_{b}.

Now suppose {b_{k}} ∈ ℓ^{1}, and let z_{0} ∈ ℂ^{N} with |z_{0}| = 1. Now there is a C depending on z_{0} such that |K_{b}(z_{0}, w)| ≤ C for all w ∈ 𝔹. On the other hand,

1 ≤ K_{q}(w, w) = ∑_{k=0}^{∞} b_{k} |w|^{2} ≤ C (w ∈ 𝔹)

since b_{k} > 0 with b_{0} = 1. Then |K_{b}(z_{0}, w)| ≤ C\sqrt{K_{b}(w, w)} = C ∥K_{b}(·, w)∥_{q} for all w ∈ 𝔹. Using the fact that the functions K_{q}(·, w) are dense in F_{q}, we conclude that z_{0} is a virtual point of F_{b}. □

The proof of the next result is clear by Remark 4.3. The case q = −N is in [CG, p. 128].

Corollary 5.3. The Dirichlet spaces F_{q} are analytic Hilbert modules if and only if q ≥ −(1 + N).

6. Shift Operators

The importance of shift operators acting on function spaces cannot be overemphasized in operator theory. Here we develop the essential properties of the multivariable shift operator on the spaces F_{b} and F_{q} realized as multiplications by the coordinate variables of ℂ^{N}.

Notation 6.1. If T = (T_{1}, …, T_{N}) is an N-tuple of operators on a Hilbert space H, for convenience we write T^{∗} := (T_{1}^{∗}, …, T_{N}^{∗}) and use the short-hand notation

T T^{∗} := T_{1}T_{1}^{∗} + ⋯ + T_{N}T_{N}^{∗}  and  T^{∗}T := T_{1}^{∗}T_{1} + ⋯ + T_{N}^{∗}T_{N}.

Warning 6.2. In many other contexts, T^{∗}T represents a matrix of operators, but we do not mean that. What is meant by T^{∗}T here is just the single operator defined by the sum in Notation 6.1.

Definition 6.3. Let H be a Hilbert space containing ℂ[z] and let S_{j} : H → H be the operator of multiplication by z_{j}, j = 1, …, N. The operator S = (S_{1}, …, S_{N}) is called the (forward, unilateral) shift operator on H. When H = F_{b}, we write S_{b} for S and call it the b-shift, and when H = F_{q}, we write S_{q} for S and call it the q-shift.

The operator of multiplication M_{p} by p ∈ ℂ[z] defined on H can also be expressed as the operator p(S) = p(S_{1}, …, S_{N}).

Let S_{b_{j}}^{∗} be the adjoint of S_{b_{j}}. Then S_{b_{j}}z^{α} = z^{α+e_{j}}, and S_{b_{j}}^{∗}z^{α} = C_{j} z^{α−e_{j}} for α_{j} > 0 by the uniqueness of the adjoint. Clearly S_{b_{j}}^{∗}z^{α} = 0 if α_{j} = 0 and so S_{b_{j}}^{∗}1 = 0. To find C_{j}, we compute

[S_{b_{j}}^{∗}z^{α}, z^{α−e_{j}}]_{b} = [z^{α}, S_{b_{j}}z^{α−e_{j}}]_{b} = [z^{α}, z^{α}]_{b} = ∥z^{α}∥_{b}^{2}
= [C_{j} z^{α−e_{j}}, z^{α−e_{j}}]_{b} = C_{j} ∥z^{α−e_{j}}∥_{b}^{2}.

Hence by (13),

(24)  S_{b_{j}}^{∗}z^{α} = C_{j} z^{α−e_{j}} = \frac{∥z^{α}∥_{b}^{2}}{∥z^{α−e_{j}}∥_{b}^{2}} z^{α−e_{j}} = \frac{α_{j}}{|α|} \frac{b_{|α|−1}}{b_{|α|}} z^{α−e_{j}}  (|α| > 0).

Apparently the S_{b_{j}} commute with each other, and so do the S_{b_{j}}^{∗}, but S_{b_{j}} does not commute with S_{b_{j}}^{∗}. Then using Notation 6.1, S_{b}S_{b}^{∗}1 = 0,

S_{b}S_{b}^{∗}z^{α} = \frac{b_{|α|−1}}{b_{|α|}} z^{α} (|α| > 0), and S_{b}^{∗}S_{b}z^{α} = \frac{N + |α|}{1 + |α|} \frac{b_{|α|}}{b_{|α|+1}} z^{α} (|α| ≥ 0).

Note that the coefficients of z^{α} in S_{b}S_{b}^{∗} and S_{b}^{∗}S_{b} depend on |α| and not on the components of α. Thus if f ∈ F_{b} is given by its homogeneous expansion (5), then

(25)  S_{b}S_{b}^{∗}f = ∑_{k=1}^{∞} \frac{b_{k−1}}{b_{k}} f_{k}  and  S_{b}^{∗}S_{b}f = ∑_{k=0}^{∞} \frac{N + k}{1 + k} \frac{b_{k}}{b_{k+1}} f_{k}.

Theorem 3.5 immediately implies the following. The extra conditions on b are not vacuous by [R, Example 3.35].

Proposition 6.4. The operators S_{b}S_{b}^{∗} and S_{b}^{∗}S_{b} are positive. They are bounded if and only the sequence {b_{k}/b_{k+1}} is bounded, in which case S_{b_{j}} and S_{b} are also bounded along with their adjoints. Also ∥S_{b}S_{b}^{∗}∥ = sup{b_{k}/b_{k+1}} and a counterpart for ∥S_{b}^{∗}S_{b}∥ holds. Further, S_{b}S_{b}^{∗} and S_{b}^{∗}S_{b} are compact if and only if b_{k}/b_{k+1} → 0 as k → ∞, which also implies the compactness of S_{b_{j}}^{∗} and S_{b}^{∗}. Moreover, S_{b}S_{b}^{∗} and S_{b}^{∗}S_{b} belong to C^{p}(F_{b}) if and only if {(b_{k}/b_{k+1}) k^{(N−1)/p}} ∈ ℓ^{p}. Finally, S_{b}^{∗}S_{b} is invertible, but S_{b}S_{b}^{∗} is not.

Continuing,

(I − S_{b}S_{b}^{∗})f = f_{0} + ∑_{k=1}^{∞}(1 − \frac{b_{k−1}}{b_{k}} ) f_{k}, (I − S_{b}^{∗}S_{b})f = ∑_{k=0}^{∞}(1 − \frac{N + k}{1 + k} \frac{b_{k}}{b_{k+1}} ) f_{k}.

So unless {b_{k}} is a constant sequence, the operator I − S_{b}S_{b}^{∗} is not the projection onto the one-dimensional space of constants in F_{b}. The coefficients are all positive if and only if {b_{k}} is a increasing sequence. Even if b_{m} < b_{m−1} for one m > 0, substituting f(z) = z_{1}^{m} to the above equation yields [(I − S_{b}S_{b}^{∗})f, f]_{b} < 0. In such a case,

λ = \frac{b_{m} − b_{m−1}}{b_{m}} < 0

is a negative eigenvalue with eigenvector z_{1}^{m} of I − S_{b}S_{b}^{∗}. Now the following results are also direct implications of Theorem 3.5.

Proposition 6.5. The operator I − S_{b}S_{b}^{∗} is positive if and only if {b_{k}} is an increasing sequence. Moreover, I − S_{b}S_{b}^{∗} = 1 ⊗ 1 if and only if {b_{k}} is a constant sequence with the constant equal to b_{0} = 1.

Proposition 6.6. The operator I −S_{b}^{∗}S_{b} is positive if and only if {b_{k}} is a sequence that increases initially fast in the sense that (1+k) b_{k+1} ≥ (N +k) b_{k} which reduces to increasing for N = 1.

Definition 6.7. A row contraction on a Hilbert space H is a commuting N-tuple of operators T = (T_{1}, …, T_{N}) on H satisfying

∥T_{1}u_{1} + ⋯ + T_{N}u_{N}∥_{H}^{2} ≤ ∥u_{1}∥_{H}^{2} + ⋯ + ∥u_{N}∥_{H}^{2}

for all u_{1}, …, u_{N} ∈ H. The norm of a Hilbert space containing polynomials in N complex variables is called contractive if the associated shift is a row contraction and the constants are orthogonal to all polynomials vanishing at 0.

Note that if T is a row contraction, then any individual T_{j} is a contraction, but the converse of this statement need not be true. By [Ar3, Remark 3.2], T is a row contraction if and only if I − T T^{∗} is a positive operator.

Proposition 6.8. The b-shift S_{b} is a row contraction and ∥ · ∥_{b} is a contractive norm if and only if {b_{k}} is increasing. In such a case, ∥ · ∥_{b} ≤ ∥ · ∥_{A}. The adjoint S_{b}^{∗} of the b-shift is a row contraction if and only if {b_{k}} is increasing initially fast in the sense described in Proposition 6.6.

Proof. For the first statement, we combine the previous paragraph with Propositions 6.5 and 3.4. For the second, we use [Ar3, Theorem 4.3], which also implies b_{k} ≥ 1 = b_{0}, which is obvious for an increasing {b_{k}}. For the third, we use Proposition 6.6 instead. □

We use Theorem 3.5 one more time to conclude the following.

Proposition 6.9. The operator I − S_{b}S_{b}^{∗} is compact if and only if b_{k}/b_{k+1} → 1 as k → ∞, and I − S_{b}S_{b}^{∗} ∈ C^{p}(F_{b}) if and only if {(1 − b_{k}/b_{k+1}) k^{(N−1)/p}} ∈ ℓ^{p}. Similar results hold for I − S_{b}^{∗}S_{b}.

Specializing to H = F_{q}, by Definition 4.1 and (24), we have S_{q_{j}}^{∗}1 = 0, and

(26)  S_{q_{j}}^{∗}z^{α} = \begin{cases}\frac{α_{j}}{N + q + |α|} z^{α−e_{j}}, & if q > −(1 + N), \\ \frac{(−(N + q) + |α|) α_{j}}{|α|^{2}} z^{α−e_{j}}, & if q ≤ −(1 + N), \end{cases}

for |α| > 0, meaning that S_{q_{j}}^{∗}z^{α} = 0 if α_{j} = 0. Also

S_{q}S_{q}^{∗}z^{α} = \begin{cases}\frac{|α|}{N + q + |α|} z^{α}, & if q > −(1 + N), \\ \frac{−(N + q) + |α|}{|α|} z^{α}, & if q ≤ −(1 + N), \end{cases}

S_{q}^{∗}S_{q}z^{α} = \begin{cases}\frac{N + |α|}{1 + N + q + |α|} z^{α}, & if q > −(1 + N), \\ \frac{(1 − (N + q) + |α|) (N + |α|)}{(1 + |α|)^{2}} z^{α}, & if q ≤ −(1 + N), \end{cases}

and

(27)  S_{q}S_{q}^{∗}f = \begin{cases}∑_{k=1}^{∞} \frac{k}{N + q + k} f_{k}, & if q > −(1 + N), \\ ∑_{k=1}^{∞} \frac{−(N + q) + k}{k} f_{k}, & if q ≤ −(1 + N), \end{cases}

(28)  S_{q}^{∗}S_{q}f = \begin{cases}∑_{k=0}^{∞} \frac{N + k}{1 + N + q + k} f_{k}, & if q > −(1 + N), \\ ∑_{k=0}^{∞} \frac{(1 − (N + q) + k) (N + k)}{(1 + k)^{2}} f_{k}, & if q ≤ −(1 + N). \end{cases}

The coefficient of f_{k} in S_{q}S_{q}^{∗}f is increasing in k if q ≥ −N and decreasing otherwise. The coefficient of f_{k} in S_{q}^{∗}S_{q}f is increasing in k if q ≥ −1 and decreasing otherwise. Thus

(29)  ∥S_{q}S_{q}^{∗}∥ = \begin{cases}1, & if q ≥ −N, \\ 1/(1 + N + q), & if − (1 + N) < q < −N, \\ 1 − (N + q), & if q ≤ −(1 + N), \end{cases}

and

(30)  ∥S_{q}^{∗}S_{q}∥ = \begin{cases}1, & if q ≥ −1, \\ N/(1 + N + q), & if − (1 + N) < q < −1, \\ N(1 − (N + q)), & if q ≤ −(1 + N). \end{cases}

In particular, every S_{q_{j}} and every S_{q} are bounded operators along with their adjoints. The results for S_{q}S_{q}^{∗} appear in [AZ, Corollary 3.7 and Theorem 4.1] for Dirichlet spaces on bounded symmetric domains restricted to the cases counterpart to q > −(1 + N). The result for S_{−N}^{∗}S_{−N} is in [Ar3, Proposition 5.3]. Hence lim_{q→−∞} ∥S_{q}S_{q}^{∗}∥ = lim_{q→−∞} ∥S_{q}^{∗}S_{q}∥ = ∞. The more interesting observation is that

lim_{q→−(1+N)^{+}} ∥S_{q}S_{q}^{∗}∥ = lim_{q→−(1+N)^{+}} ∥S_{q}^{∗}S_{q}∥ = ∞ while ∥S_{−(1+N)}S_{−(1+N)}^{∗}∥ = 2 and

∥S_{−(1+N)}^{∗}S_{−(1+N)}∥ = 2N.

It is not difficult to compute the precise norms of the individual S_{q_{j}}, and in fact of any S_{q}^{β}. First

∥S_{q_{j}}f∥_{q}^{2} = ∑_{α} |f_{α}|^{2} ∥z^{α+e_{j}}∥_{q}^{2} = ∑_{α} |f_{α}|^{2} ∥z^{α}∥_{q}^{2} μ_{α}(j),

where

μ_{α}(j) = \begin{cases}\frac{1 + α_{j}}{1 + N + q + |α|}, & if q > −(1 + N), \\ \frac{1 − (N + q) + |α|}{1 + |α|} \frac{1 + α_{j}}{1 + |α|}, & if q ≤ −(1 + N) \end{cases}

by (18). For q ≥ −N, always μ_{α}(j) ≤ 1, we consider f(z) = z_{j}^{α_{j}}, and let α_{j} → ∞. For q < −N, μ_{α}(j) is largest when α = (0, …, 0), and we consider f(z) = 1. Then we obtain

∥S_{q_{j}}∥ = \begin{cases}1, & if q ≥ −N, \\ 1/\sqrt{1 + N + q}, & if − (1 + N) < q < −N, \\ \sqrt{1 − (N + q)}, & if q ≤ −(1 + N), \end{cases}

where the critical values of q are those of (29). Thus the individual shifts on F_{q} are contractions if and only if q ≥ −N, and this range includes the Drury-Arveson space. Again lim_{q→−(1+N)^{+}} ∥S_{q_{j}}∥ = ∞ while ∥S_{−(1+N)_{j}}∥ = \sqrt{2}. Letting p(z) = z^{β} and arguing similarly, we also see that

∥S_{q}^{β}∥ = \begin{cases}1, & if q ≥ −N, \\ \begin{matrix}√ & \overline{β!} \\ √ & \overline{(1 + N + q)_{|β|}} \end{matrix} , & if − (1 + N) < q < −N, \\ \frac{\sqrt{β! (1 − (N + q))_{|β|}}}{|β|!} , & if q ≤ −(1 + N). \end{cases}

From Proposition 6.4, we obtain our next result.

Corollary 6.10. The operators S_{q}S_{q}^{∗} and S_{q}^{∗}S_{q} are never compact.

We also have

(I − S_{q}S_{q}^{∗})f = \begin{cases}f_{0} + ∑_{k=1}^{∞} \frac{N + q}{N + q + k} f_{k}, & if q > −(1 + N), \\ f_{0} + ∑_{k=1}^{∞} \frac{N + q}{k} f_{k}, & if q ≤ −(1 + N). \end{cases}

and

(31) (I − S_{q}^{∗}S_{q})f = \begin{cases}∑_{k=0}^{∞} \frac{1 + q}{1 + N + q + k} f_{k}, & if q > −(1 + N), \\ ∑_{k=0}^{∞} \frac{1 + N(−1 + N + q) + (1 + q)k}{(1 + k)^{2}} f_{k}, & if q ≤ −(1 + N). \end{cases}

The following is now clear by Remark 4.3 and Proposition 6.5.

Corollary 6.11. The operator I − S_{q}S_{q}^{∗} is positive, equivalently S_{q} is a row contraction, equivalently ∥ · ∥_{q} is a contractive norm, if and only if q ≥ −N. In such a case, ∥ · ∥_{q} ≤ ∥ · ∥_{−N}. Further, I − S_{q}S_{q}^{∗} is of finite rank and in such a case equals 1 ⊗ 1 if and only if q = −N, that is, only for the Drury-Arveson space F_{−N}. If q ≠ −N, then I − S_{q}S_{q}^{∗} is invertible.

In fact, for q > −N and |α| > 0, ∥z^{α}∥_{q} < ∥z^{α}∥_{−N} as can be seen from (18). For q < −N, the explicit forms of ∥z^{α}∥_{q}^{2} in (18) show that ∥z^{α}∥_{q} > ∥z^{α}∥_{−N} for |α| > 0, and hence for all f ∈ F_{q} except for constants. This observation combined with Corollary 4.5 implies that within the scale of spaces F_{q}, the Drury-Arveson space A (q = −N) is the smallest one with a contractive norm. This should be compared to the fact that within the same scale, this space is the largest one with a complete Nevanlinna-Pick kernel; see [AM, Theorem 7.33 and Corollary 7.41].

Definition 6.12. An N-tuple T = (T_{1}, …, T_{N}) of operators is called a spherical isometry if I − T^{∗}T = 0, a spherical contraction if I − T^{∗}T ≥ 0, and a spherical expansion if I − T^{∗}T ≤ 0, where we use Notation 6.1.

Corollary 6.13. The N-tuple S_{q} is a spherical contraction if and only if q ≥ −1, and it is a spherical expansion if and only if q ≤ −1. It is a spherical isometry if and only if q = −1, that is, only on the Hardy space F_{−1}. If q ≠ −1, then I − S_{q}^{∗}S_{q} is invertible.

Proof. Consider (31). If q > −(1 + N), positivity of I − S_{q}^{∗}S_{q} holds only for q ≥ −1. If q ≤ −(1 + N), then −1 + N + q ≤ −2 and the coefficient of f_{0} is 1 + N(−1 + N + q) ≤ 1 − 2N < 0; so there is no positivity for q ≤ −(1 + N). □

Corollary 6.14. Both operators I − S_{q}S_{q}^{∗} and I − S_{q}^{∗}S_{q} are always compact, and they belong to C^{p}(F_{q}) if and only if {k^{(N−1−p)/p}} ∈ ℓ^{p} if and only if p > N.

The condition p > N in Corollary 6.14 and Proposition 4.4 is the same, because the operators (I + R)^{−1}, I − S_{q}S_{q}^{∗}, and I − S_{q}^{∗}S_{q} have their coefficient multiplier sequence ∼ 1/k.

The formulas for S_{q}S_{q}^{∗} and S_{q}^{∗}S_{q} bear a strong resemblance to the definitions of the D_{s}^{t} and R, and indeed we are able to write them in terms of simple combinations of our radial differential operators. It is easy to check that

S_{q}S_{q}^{∗} = \begin{cases}\frac{1}{N + q} D_{q}^{−1}R, & if q > −N, \\ −(N + q) (R′)^{−1}D_{q}^{1}, & if q ≤ −(2 + N), \end{cases}

= \begin{cases}(R + (N + q)I)^{−1}R, & if q > −(1 + N), \\ (R − (N + q)I) (R′)^{−1}, & if q ≤ −(1 + N), \end{cases}

where R′ represents the restriction of R to F_{q}/ℂ on which it is invertible with inverse

(32)  (R′)^{−1} = (1 ⊗ 1 + R)^{−1} − 1 ⊗ 1,

and

S_{q}^{∗}S_{q} = \begin{cases}\frac{N}{1 + N + q} D_{−1}^{1}D_{1+q}^{−1}, & if q > −(1 + N), \\ N (1 − (N + q)) D_{−1}^{1}D_{−1+q}^{1}(D_{−N}^{−1})^{2}, & if q ≤ −(1 + N). \end{cases}

A formula for (S_{q}^{∗}S_{q})^{−1} can be written using (20), but we have no use for it. Similar formulas can be tried for the other operators as well.

7. von Neumann Inequalities

We are now ready to prove a von Neumann inequality with respect to every weighted symmetric Fock space F_{b} for row contractions on arbitrary Hilbert spaces. It turns out that among the scale of spaces F_{q}, the inequality with respect to the Drury-Arveson space A is the simplest one, though not always the sharpest one.

Definition 7.1. Let T = (T_{1}, …, T_{N}) be a row contraction on a Hilbert space H. Its defect operator is Δ_{T} = (I − T T^{∗})^{1/2}, which is the unique positive square root.

Clearly Δ_{S_{b}}1 = 1 for any b. The computations in Section 6 show that

Δ_{S_{b}}f = f_{0} + ∑_{k=1}^{∞} \sqrt{1 − \frac{b_{k−1}}{b_{k}}} f_{k}  (\{b_{k}\} is increasing),

Δ_{S_{q}}f = f_{0} + ∑_{k=1}^{∞} \sqrt{\frac{N + q}{N + q + k}} f_{k}  (q ≥ −N).

An operator tuple T = (T_{1}, …, T_{N}) on H gives rise to a map J_{T} : B(H) → B(H) defined by

J_{T} (A) := T_{1}AT_{1}^{∗} + ⋯ + T_{N}AT_{N}^{∗} (A ∈ B(H)).

If T is a row contraction, then I ≥ J_{T} (I) = T T^{∗} ≥ J_{T}^{2} (I) ≥ ⋯ ≥ 0 using Notation 6.1. Thus lim_{n→∞} J_{T}^{n}(I) = T_{∞} exists in the strong operator topology and satisfies 0 ≤ T_{∞} ≤ I.

Definition 7.2. If T_{∞} = 0 for a row contraction T on a Hilbert space H, then T is called null (or pure).

If T T^{∗} ≤ rI with 0 < r < 1, then T is clearly null.

When H = F_{b} and T = S_{b}, the b-shift, a straightforward computation using (25) repeatedly shows that

J_{S_{b}}^{m}(I) z^{α} = \frac{b_{|α|−m}}{b_{|α|}} z^{α} (|α| ≥ m)  and  J_{S_{b}}^{m}(I)f = ∑_{k=m}^{∞} \frac{b_{k−m}}{b_{k}} f_{k}

for f ∈ F_{b} with the homogeneous expansion (5); also J_{S_{b}}^{m}(I)p = 0 if p is a polynomial of degree less than m. In particular,

J_{S_{q}}^{m}(I)f = \begin{cases}∑_{k=m}^{∞} \frac{(1 + k − m)_{m}}{(1 + N + q + k − m)_{m}} f_{k}, & if q > −(1 + N), \\ ∑_{k=m}^{∞} \frac{(1 − (N + q) + k − m)_{m}}{(1 + k − m)_{m}} f_{k}, & if q ≤ −(1 + N). \end{cases}

If S_{b} is a row contraction which happens if and only if {b_{k}} is increasing by Proposition 6.8, then the coefficients of the f_{k} in J_{S_{b}}^{m}(I)f are all ≤ 1, and

(33)  ∥J_{S_{b}}^{m}(I)f∥_{b}^{2} = ∑_{k=m}^{∞} \frac{b_{k−m}^{2}}{b_{k}^{2}} ∥f_{k}∥_{k}^{2} ≤ ∑_{k=m}^{∞} ∥f_{k}∥_{k}^{2} → 0  as  m → ∞,

because it is the tail end of the series in (6). Thus (S_{b})_{∞} = 0 for S_{b} a row contraction. In particular, this holds for S_{q} when q ≥ −N.

On the other hand, by first replacing k − m by k, it is easy to see that

(34)  ∥J_{S_{q}}^{m}(I)f∥_{q}^{2} ∼ m^{−2(N+q)} ∑_{k=m}^{∞} ∥f_{k}∥_{q}^{2}.

For q < −N, the power on m is strictly positive. For such q, define

f̃(z) = ∑_{k=1}^{∞} \frac{1}{k^{(1−2(N+q))/2}} z_{1}^{k}.

Then by (18),

∥f̃∥_{q}^{2} ∼ ∑_{k=1}^{∞} \frac{1}{k^{1−2(N+q)}} \frac{1}{k^{N+q}} < ∞,

that is, f ∈ F_{q}. But

∑_{k=m}^{∞} ∥f̃_{k}∥_{q}^{2} = ∑_{k=m}^{∞} \frac{1}{k^{1−(N+q)}} ∼ ∫_{m}^{∞} \frac{dx}{x^{1−(N+q)}} ∼ \frac{1}{m^{−(N+q)}} .

Thus ∥J_{S_{q}}^{m}(I)f̃∥_{q}^{2} ∼ m^{−(N+q)} → ∞ as m → ∞.

The following result emerges by collecting the pieces together.

Proposition 7.3. The b-shift S_{b} is null whenever it is a row contraction. In particular, this holds for every q-shift S_{q} with q ≥ −N. Conversely, S_{q} is not null for q < −N.

This fact has been noted for q > −(1 + N) in [AZ, Lemma 3.8], and for shifts of a different kind in [MV, Lemma 7].

We make a digression to note a consequence of the above computations.

Theorem 7.4. If {b_{k}} is increasing, then (S_{b_{j}}^{∗})^{m} → 0 as m → ∞ in the strong operator topology. Further, (S_{q_{j}}^{∗})^{m} → 0 as m → ∞ in the strong operator topology if and only if q ≥ −N.

Proof. As m → ∞, (S_{b_{j}}^{∗})^{m} → 0 in the strong operator topology if and only if ∥(S_{b_{j}}^{∗})^{m}f∥_{b}^{2} = [f, S_{b_{j}}^{m}(S_{b_{j}}^{∗})^{m}f]_{b} → 0 for every f ∈ F_{b}. But

[f, S_{b_{j}}^{m}(S_{b_{j}}^{∗})^{m}f]_{b} = ∑_{k=m}^{∞} ∑_{|α|=k} \frac{(1 + α_{j} − m)_{m}}{(1 + k − m)_{m}} \frac{b_{k−m}}{b_{k}} |f_{α}|^{2} ∥z^{α}∥_{b}^{2}.

The first fraction is always ≤ 1. If {b_{k}} is increasing, then the second fraction is also ≤ 1 and [f, S_{b_{j}}^{m}(S_{b_{j}}^{∗})^{m}f]_{b} → 0 as m → ∞ for all f ∈ F_{b} as in (33). The convergence for q ≥ −N also follows immediately.

For q < −N, the f̃ above can be used again. For this function, α_{1} = k and the first fraction in [f̃, S_{q_{1}}^{m}(S_{q_{1}}^{∗})^{m}f̃]_{q} equals 1. Then the only difference of

[f̃, S_{q_{1}}^{m}(S_{q_{1}}^{∗})^{m}f̃]_{q} from ∥J_{S_{q}}^{m}(I)f̃∥_{q}^{2} in (34) is that the power on m now is −(N + q).

After estimating the norm of the tail end of f̃, we obtain [f̃, S_{q_{1}}^{m}(S_{q_{1}}^{∗})^{m}f̃]_{q} ∼ 1,

which does not tend to 0 as m → ∞. For j ≠ 1, just replace z_{1} by z_{j} in f̃. □

Back to our main route, we next transform row contractions to any F_{b} building on the treatment in [CG, Theorem 6.1.4].

Theorem 7.5. Let T = (T_{1}, …, T_{N}) be a row contraction on a Hilbert space H. For any b = \{b_{k}\}, there exists a unique bounded operator L_{b} : F_{b} ⊗ \overline{Δ_{T} H} → H which is a contraction satisfying

(35)  L_{b}(z^{α} ⊗ v) = \frac{1}{\sqrt{b_{|α|}}} T^{α}Δ_{T} v  (v ∈ H, α ∈ ℕ^{N}).

In particular, L_{b}(1 ⊗ v) = Δ_{T} v for v ∈ H. If T is null, then L_{b} is a coisometry, that is, L_{b}L_{b}^{∗} = I_{H}.

Proof. Uniqueness follows from the fact that members of F_{b} are holomorphic functions on 𝔹 with Taylor expansions in terms of {z^{α}}.

Set H_{0} = \overline{Δ_{T} H} and note that H_{0} is a subspace of H. For each multi-index α, put k = |α|, and for u ∈ H, let

y_{b_{α}} = \frac{k!}{α!} \sqrt{b_{k}} z^{α} ⊗ Δ_{T} T^{∗α}u  and  y_{b_{k}} = ∑_{|α|=k} y_{b_{α}},

which evidently belong to E^{k} ⊗ H_{0}. In particular, y_{b_{0}} = 1 ⊗ Δ_{T} u. By (15) and (13), we have

∥y_{b_{k}}∥_{E^{k}⊗H_{0}}^{2} = ∑_{|α|=k} \frac{(k!)^{2}}{(α!)^{2}} b_{k} ∥z^{α}∥_{k}^{2} ∥Δ_{T} T^{∗α}u∥_{H}^{2} = ∑_{|α|=k} \frac{k!}{α!} [Δ_{T} T^{∗α}u, Δ_{T} T^{∗α}u]_{H}

= ∑_{|α|=k} \frac{k!}{α!} [T^{α}Δ_{T}^{2} T^{∗α}u, u]_{H} = [J_{T}^{k}(Δ_{T}^{2} )u, u]_{H}

= [J_{T}^{k}(I − J_{T} (I))u, u]_{H} = [(J_{T}^{k}(I) − J_{T}^{k+1}(I))u, u]_{H}.

Define an operator A_{b} : H → F_{b} ⊗ H_{0} by A_{b}u = (y_{b_{0}}, y_{b_{1}}, y_{b_{2}}, …). Then

∥A_{b}u∥_{F_{b}⊗H_{0}}^{2} = ∑_{k=0}^{∞} ∥y_{b_{k}}∥_{E^{k}⊗H_{0}}^{2} = ∑_{k=0}^{∞} [(J_{T}^{k}(I) − J_{T}^{k+1}(I))u, u]_{H}
= ∥u∥_{H}^{2} − [T_{∞}u, u]_{H} ≤ ∥u∥_{H}^{2} (u ∈ H).

Thus ∥A_{b}∥ ≤ 1. Equality holds and A_{b} is an isometry if T is null.

We now define L_{b} = A_{b}^{∗}. Then ∥L_{b}∥ ≤ 1, and L_{b} is a coisometry if T is null. Using (15) and (13), for u ∈ H, v ∈ H_{0}, and a multi-index α, we compute

[L_{b}(z^{α} ⊗ v), u]_{H} = [z^{α} ⊗ v, A_{b}u]_{F_{b}⊗H_{0}} = [z^{α} ⊗ v, y_{b_{α}}]_{E^{k}⊗H_{0}}

= [z^{α} ⊗ v, \frac{k!}{α!} \sqrt{b_{k}} z^{α} ⊗ Δ_{T} T^{∗α}u]_{E^{k}⊗H_{0}}

= \frac{k!}{α!} \sqrt{b_{k}} ∥z^{α}∥_{k}^{2} [v, Δ_{T} T^{∗α}u]_{H} = \frac{1}{\sqrt{b_{k}}} [T^{α}Δ_{T} v, u]_{H},

which establishes (35). □

This theorem naturally leads to von Neumann inequalities, one with respect to each F_{b}, which are new even when N = 1.

Theorem 7.6. Let T = (T_{1}, …, T_{N}) be a row contraction on a Hilbert space H.

Let a weight sequence b be chosen with the property that \{\frac{b_{k+m}}{b_{m}} \} is bounded above

by a constant K_{k} > 0 that depends only on k. And let p(z) = ∑_{k=0}^{d} p_{k}(z) be a

polynomial in N complex variables, where p_{k} is homogeneous of degree k. Then

∥p(T)∥ ≤ ∑_{k=0}^{d} \sqrt{K_{k}} ∥p_{k}(S_{b})∥,

where S_{b} is the b-shift.

Proof. By Theorem 7.5, we have a contraction L_{b} : F_{b} ⊗ H_{0} → H satisfying (35); equivalently,

L_{b}(S_{b}^{α} ⊗ I_{H_{0}})(1 ⊗ v) = \frac{1}{\sqrt{b_{k}}} T^{α}Δ_{T} v (v ∈ H_{0}, α ∈ ℕ^{N}),

where we continue to use H_{0} = \overline{Δ_{T} H} and k = |α|. Let β be another multi-index and m = |β|. Then

L_{b}(S_{b}^{α} ⊗ I_{H_{0}})(z^{β} ⊗ v) = L_{b}(z^{α+β} ⊗ v) = \frac{1}{\sqrt{b_{k+m}}} T^{α}Δ_{T} v = \sqrt{\frac{b_{m}}{b_{k+m}}} T^{α}L_{b}(z^{β} ⊗ v)

and

(36)  L_{b}(S_{b}^{α} ⊗ I_{H_{0}})(g_{k} ⊗ v) = T^{α}L_{b}(f ⊗ v),

where

f(z) = ∑_{β} f_{β} z^{β} ∈ F_{b}, g_{k}(z) = ∑_{β} \sqrt{\frac{b_{k+m}}{b_{m}}} f_{β} z^{β},

and g_{k} ∈ F_{b} by the assumption on b since ∥g_{k}∥_{F_{b}} ≤ \sqrt{K_{k}} ∥f∥_{F_{b}}.

We first consider the case T is null under which L_{b}^{∗} is an isometry. Given an arbitrary u ∈ H, let f ⊗ v = L_{b}^{∗}u. Then ∥u∥_{H} = ∥f ⊗ v∥_{F_{b}⊗H} = ∥f∥_{F_{b}} ∥v∥_{H}.

It follows that T^{α}L_{b}(f ⊗ v) = T^{α}L_{b}L_{b}^{∗}u = T^{α}u = L_{b}(S_{b}^{α} ⊗ I_{H_{0}})(g_{k} ⊗ v) and

p(T)u = L_{b} ∑_{k=0}^{d} p_{k}(S_{b})(g_{k}) ⊗ v. Thus

∥p(T)u∥_{H} = ∑_{k=0}^{d} ∥p_{k}(S_{b})g_{k}∥_{F_{b}} ∥v∥_{H} ≤ ∑_{k=0}^{d} ∥p_{k}(S_{b})∥ ∥g_{k}∥_{F_{b}} ∥v∥_{H}

≤ ∑_{k=0}^{d} \sqrt{K_{k}} ∥p_{k}(S_{b})∥ ∥f∥_{F_{b}} ∥v∥_{H} = ∑_{k=0}^{d} \sqrt{K_{k}} ∥p_{k}(S_{b})∥ ∥u∥_{H}

and ∥p(T)∥ ≤ ∑_{k=0}^{d} \sqrt{K_{k}} ∥p_{k}(S_{b})∥ when T is null.

If T is not null, for 0 < r < 1, set rT = (rT_{1}, …, rT_{N}), which is null since (rT)(rT)^{∗} ≤ r^{2}I. Applying the above to rT, we obtain the same upper bound on ∥p(rT)∥ which is independent of r or T. Letting r → 1^{−} finishes the proof. □

When F_{b} = A, the Drury-Arveson space, and only then, we have g_{k} = f for all k in the above proof. Then (36) yields T^{α} = L_{b}(S_{b}^{α} ⊗ I_{H_{0}})L_{b}^{∗} when T is null. From here the von Neumann inequality of [Dr, Theorem] and [Ar3, Theorem 8.1] is obtained, which is ∥p(T)∥ ≤ ∥p(S_{−N})∥. Clearly, the inequality of this particular case is in general sharper than those of the other spaces considered in this work.

The assumption on T in Theorem 7.6 is that I − T T^{∗} ≥ 0. In [P, Corollary 5.4], this condition is replaced by the positivity of more general polynomial expressions in T and T^{∗}, which also involves a consideration of Bergman spaces on domains determined by a similar polynomial inequality.

Remark 7.7. Let’s see that the condition on b in Theorem 7.6 is satisfied by the weight sequences b_{q} of all the Dirichlet spaces F_{q}. It is enough to check those other than the Drury-Arveson space F_{−N}. If q > −(1 + N), then

\frac{b_{k+m}(q)}{b_{m}(q)} = \frac{(1 + N + q + m)_{k}}{(1 + m)_{k}}  and  lim_{m→∞} \frac{b_{k+m}(q)}{b_{m}(q)} = 1.

In the subcase q > −N, then the ratios are all > 1 and the smallest

K_{k} = \frac{b_{k}(q)}{b_{0}(q)} = b_{k}(q) = \frac{(1 + N + q)_{k}}{k!} > 1.

In the other subcase −(1 + N) < q < −N, then the ratios are all < 1 and the smallest

K_{k} = lim_{m→∞} \frac{b_{k+m}(q)}{b_{m}(q)} = 1.

If q ≤ −(1 + N), then

\frac{b_{k+m}(q)}{b_{m}(q)} = \frac{(1 + m)_{k}}{(1 − (N + q) + m)_{k}}  and  lim_{m→∞} \frac{b_{k+m}(q)}{b_{m}(q)} = 1.

Since the ratios are all less than 1 now, the smallest K_{k} = 1 again.

8. Spectra and Commutants

We continue our investigation of the shift operators on the F_{b}. We observe that they are weighted shift operators when (and only when) N = 1. This yields a multitude of properties for them. We touch upon some that are related to their spectra. Later we determine the commutants of the shifts independently of N; they turn out to be the multiplier algebras of the F_{b} as can be expected.

All our shifts are undoubtedly one-to-one. By Proposition 3.4,

S_{b_{j}}z^{α} = \sqrt{\frac{α!}{b_{|α|} |α|!}} S_{b_{j}}O_{α}^{b}(z) = z^{α+e_{j}} = \sqrt{\frac{α! (1 + α_{j})}{b_{1+|α|} (1 + |α|)!}} O_{α+e_{j}}^{b}(z),

so

S_{b_{j}}O_{α}^{b}(z) = \sqrt{\frac{b_{|α|} (1 + α_{j})}{b_{1+|α|} (1 + |α|)}} O_{α+e_{j}}^{b}(z).

Proposition 8.1. When N = 1, the shift S_{b} is a weighted shift operator with the weight sequence

ω_{b}(k) = \sqrt{\frac{b_{k}}{b_{1+k}}} = \frac{∥z^{1+k}∥_{b}}{∥z^{k}∥_{b}} (k = 0, 1, 2, …).

For S_{q}, the weight sequence has the form

ω_{q}(k) = \begin{cases}\sqrt{\frac{1 + k}{2 + q + k}} , & if q > −2, \\ \sqrt{\frac{−q + k}{1 + k}} , & if q ≤ −2. \end{cases}

The weight sequence of the weighted shift S_{b} is not the same as the weight sequence {b_{k}} of the weighted Fock space F_{b}, although they are closely related by this proposition. When N > 1, it is clear that the shifts S_{b_{j}} and S_{q_{j}} are not weighted shifts.

Every shift is a multiplication operator. Thus by (23), if w ∈ 𝔹, then \overline{w}_{j} is an eigenvalue of S_{b_{j}} with common eigenvector K_{b}(·, w) for j = 1, …, N, that is,

(37)  S_{b_{j}}^{∗}K_{b}(·, w) = \overline{w}_{j} K_{b}(·, w).

In particular, 0 is an eigenvalue of each S_{b_{j}}^{∗} with eigenvector 1. If N = 1, we prove that points of 𝔻 are the only eigenvalues of S_{b}^{∗} under a mild condition.

Theorem 8.2. Let N = 1 and recall that Z_{b} is defined in (3).

(i) We have σ_{p}(S_{b}) = ∅ and \sqrt{Z_{b}} 𝔻 ⊂ σ_{p}(S_{b}^{∗}) ⊂ \sqrt{Z_{b}} \overline{𝔻}. In addition, all eigenspaces of S_{b}^{∗} are 1-dimensional.

(ii) If also {1/ω_{b}(k)} is bounded, then im(S_{b}) = { f ∈ F_{b} : f(0) = 0 }.

(iii) If further lim_{k→∞} ω_{b}(k) exists, then it is equal to \sqrt{Z_{b}} ≥ 1, and we have

σ(S_{b}) = \sqrt{Z_{b}} \overline{𝔻}, σ_{p}(S_{b}^{∗}) = \sqrt{Z_{b}} 𝔻, σ_{e}(S_{b}) = \sqrt{Z_{b}} ∂𝔻. (iv) With S_{q}, lim_{k→∞} ω_{q}(k) = \sqrt{Z_{q}} = 1 always exists and \{1/ω_{q}(k)\} is thus

bounded.

Proof. (i) In [Sh, p. 70], the quantity r_{2} is defined. In our case, it is

r_{2} = lim inf_{k→∞} (ω_{b}(0) ⋯ ω_{b}(k − 1))^{1/k} = \sqrt{lim inf_{k→∞} \frac{1}{b_{k}^{1/k}}} = \frac{1}{\sqrt{lim sup_{k→∞} b_{k}^{1/k}}} = \sqrt{Z_{b}}

using Proposition 8.1. The claims now follow from [Sh, Theorem 8].

(ii) Without any condition on weight sequences, clearly im(S_{b}) is a subset of { f ∈ F_{b} : f(0) = 0 } which is closed since point evaluations are continuous on

F_{b} by Theorem 3.2. Conversely, if f(z) = ∑_{k=1}^{∞} f_{k}z^{k} ∈ { f ∈ F_{b} : f(0) = 0 }, then

f(z) = z g(z) for some g ∈ H(𝔻). But

∥g∥_{F_{b}} = ∑_{k=1}^{∞} \frac{|f_{k}|^{2}}{b_{k−1}} = ∑_{k=1}^{∞} \frac{|f_{k}|^{2}}{b_{k}} \frac{b_{k}}{b_{k−1}} < ∞

by the condition on {1/ω_{b}(k)}. This shows g ∈ F_{b} and f ∈ im(S_{b}).

(iii) We have

(38)  lim_{k→∞} ω_{b}(k) = \sqrt{lim_{k→∞} \frac{b_{k}}{b_{1+k}}} = \sqrt{\frac{1}{lim_{k→∞} b_{k}^{1/k}}} = \sqrt{Z_{b}} ≥ 1

by [R, Theorem 3.37] and Proposition 8.1. Then the claim about σ(S_{b}) follows from [Sh, Proposition 15].

Next we show that \sqrt{Z_{b}} e^{iθ} is not an eigenvalue of S_{b}^{∗}. If it were, we would have

an associated eigenvector f(z) = ∑_{k=0}^{∞} f_{k}z^{k} ∈ F_{b}. Writing down what this means in

detail and solving for f_{k} would give f_{k} = \sqrt{Z_{b}} e^{ikθ} b_{k} f_{0}. We need ∑_{k=0}^{∞} Z_{b}^{k} b_{k} < ∞

by (14) since f ∈ F_{b}. But Z_{b}^{k} ∼ b_{k}^{−1} by (38) and such an f would never be in F_{b}. This proves the claim about σ_{p}(S_{b}^{∗}).

Let S̃ be the weighted shift on F_{b} with the constant weight sequence {1, 1, …}. Then S_{b} − \sqrt{Z_{b}} S̃ is a weighted shift whose weight sequence converges to 0, which implies that it is compact by [Sh, Proposition 4]. So S_{b} and \sqrt{Z_{b}} S̃ are compact perturbations of each other and have the same essential spectrum. But S̃ acts just like the unilateral shift on the Hardy space H^{2} and thus σ(S̃) = ∂𝔻. This proves the claim about σ_{e}(S_{b}).

(iv) That the limit of the weight sequence is 1 is clear from Proposition 8.1. Then we have σ(S_{q}) = \overline{𝔻}, σ_{p}(S_{q}) = ∅, σ_{p}(S_{q}^{∗}) = 𝔻, and σ_{e}(S_{q}) = ∂𝔻. □

We turn to a slightly different topic now. When N = 1, the commutant {S_{b}}′ is the multiplier algebra M(F_{b}) by [Sh, Theorem 3 (b)], because S_{b} is a weighted shift. The starting point of such results is [Sa, Theorem 1], which deals with {S_{q}}′ with q = −1. In fact, a little more generally, {S_{q}}′ = H^{∞} for all q ≥ −1 with equal norms by the remarks early in Section 5. The more interesting question is what would the commutant {S_{b_{1}}, …, S_{b_{N}} }′ be when N > 1 since none of the operators S_{b_{j}} is a weighted shift. It is clear that if g ∈ M(F_{b}), then M_{g} commutes with S_{b_{1}}, …, S_{b_{N}} since they are all multiplication operators, that is, M(F_{b}) ⊂ {S_{b_{1}}, …, S_{b_{N}} }′. Again there are no others although the proof is more involved.

Lemma 8.3. Let λ = (λ_{1}, …, λ_{N}) ∈ 𝔹. If f ∈ F_{b} is a common eigenvector of S_{b_{1}}^{∗}, …, S_{b_{N}}^{∗} associated to the eigenvalues λ_{1}, …, λ_{N}, respectively, then f is unique

and f(z) = f_{(0,...,0)} K_{b}(z, \overline{λ}). Equivalently, dim ⋂_{j=1}^{N} ker(S_{b_{j}}^{∗} − λ_{j}I) = 1.

Proof. By (24), the hypothesis is for j = 1, …, N and z ∈ 𝔹 that

S_{b_{j}}^{∗}f(z) = ∑_{|α|>0} f_{α} \frac{α_{j}}{|α|} \frac{b_{|α|−1}}{b_{|α|}} z^{α−e_{j}} = ∑_{β} f_{β+e_{j}} \frac{1 + β_{j}}{1 + |β|} \frac{b_{|β|}}{b_{1+|β|}} z^{β} = ∑_{β} λ_{j} f_{β} z^{β}.

Equating the coefficients of z^{β} on the two sides and reverting to α yields

f_{α} = λ_{j} \frac{|α|}{α_{j}} \frac{b_{|α|}}{b_{|α|−1}} f_{α−e_{j}} (j = 1, …, N, α_{j} ≠ 0).

We can reach any f_{α} from f_{(0,...,0)}, because if α_{j} ≠ 0, we use the jth formula repeatedly. This allows us to construct f by determining all the f_{α} in terms of f_{(0,...,0)}. But a particular f_{α} can be obtained using the above formulas in different orders. These applications do not give a different f_{α} from each other, because applying the jth formula is like applying S_{b_{j}}, and we know the S_{b_{j}} commute with each other. Thus f is unique up to a multiplicative constant f_{(0,...,0)}.

So we take one path from (0, …, 0) to α = (α_{1}, …, α_{N}) and compute

f_{α} = λ_{1} \frac{|α|}{α_{1}} \frac{b_{|α|}}{b_{|α|−1}} f_{α−e_{1}} = λ_{1}^{2} \frac{|α| (|α| − 1)}{α_{1}(α_{1} − 1)} \frac{b_{|α|}}{b_{|α|−1}} \frac{b_{|α|−1}}{b_{|α|−2}} f_{α−2e_{1}} = ⋯

= λ_{1}^{α_{1}} \frac{|α| (|α| − 1) ⋯ (|α| − α_{1} + 1)}{α_{1}!} \frac{b_{|α|}}{b_{|α|−α_{1}}} f_{α−α_{1}e_{1}} = ⋯

= λ_{1}^{α_{1}} λ_{2}^{α_{2}} \frac{|α| (|α| − 1) ⋯ (|α| − α_{1} − α_{2} + 1)}{α_{1}! α_{2}!} \frac{b_{|α|}}{b_{|α|−α_{1}−α_{2}}} f_{α−α_{1}e_{1}−α_{2}e_{2}} = ⋯

= λ_{1}^{α_{1}} λ_{2}^{α_{2}} ⋯ λ_{N}^{α_{N}} \frac{|α|!}{α_{1}! α_{2}! ⋯ α_{N}!} \frac{b_{|α|}}{b_{0}} f_{(0,...,0)} = λ^{α} \frac{|α|!}{α!} b_{|α|} f_{(0,...,0)}.

Therefore

f(z) = ∑_{α} f_{α} z^{α} = ∑_{α} λ^{α} \frac{|α|!}{α!} b_{|α|} f_{(0,...,0)} z^{α} = f_{(0,...,0)} ∑_{k=0}^{∞} b_{k} ∑_{|α|=k} \frac{k!}{α!} λ^{α} z^{α}

= f_{(0,...,0)} ∑_{k=0}^{∞} b_{k} ⟨z, \overline{λ}⟩^{k} = f_{(0,...,0)} K_{b}(z, \overline{λ}) (z ∈ 𝔹)

by (7). □

Pieces of the proofs of the remaining two theorems in this section are rather standard. But it is not possible to find in the literature what we need in full, so we provide the details for completeness. We abbreviate {S_{b_{1}}, …, S_{b_{N}} } to {S_{b}}.

Theorem 8.4. The commutant {S_{b}}′ equals M(F_{b}).

Proof. The proof rests on the fact that the intersection of the eigenspaces of the S_{b_{j}}^{∗} is one-dimensional. Suppose T ∈ {S_{b}}′ so that T S_{b_{j}} = S_{b_{j}}T for j = 1, …, N. Then T^{∗}S_{b_{j}}^{∗} = S_{b_{j}}^{∗}T^{∗} as well. This shows that T^{∗}1 is a constant since S_{b_{j}}^{∗}1 = 0. Let w = (w_{1}, …, w_{N}) ∈ 𝔹. By (37),

T^{∗}K_{b}(·, w) = \frac{1}{\overline{w}_{j}} T^{∗}S_{b_{j}}^{∗}K_{b}(·, w) = \frac{1}{\overline{w}_{j}} S_{b_{j}}^{∗}T^{∗}K_{b}(·, w)

for w_{j} ≠ 0. Then S_{b_{j}}^{∗}(T^{∗}K_{b}(·, w)) = \overline{w}_{j} (T^{∗}K_{b}(·, w)), which is true if w_{j} = 0 also since then the series expansion of K_{b}(·, w) contains no z_{j}. So T^{∗}K_{b}(·, w) is a common eigenvector of the S_{b_{j}}^{∗} with the eigenvalues \overline{w}_{j}. Lemma 8.3 implies the existence of a constant \overline{g(w)} such that T^{∗}K_{b}(z, w) = \overline{g(w)} K_{b}(z, w). This means that \overline{g(w)} ∈ σ(T^{∗}), hence g(w) ∈ σ(T), and thus |g(w)| ≤ ∥T ∥; that is, g is a bounded function of w. Next, the computation

g(w) = g(w) [1, K_{b}(·, w)]_{b} = [1, \overline{g(w)} K_{b}(·, w)]_{b}
= [1, T^{∗}K_{b}(·, w)]_{b} = [T 1, K_{b}(·, w)]_{b} = T 1(w)

yields that g = T 1 ∈ F_{b}. Finally, for f ∈ F_{b}, the reverse computation

T f(w) = [T f, K_{b}(·, w)]_{b} = [f, T^{∗}K_{b}(·, w)]_{b}
= [f, \overline{g(w)} K_{b}(·, w)]_{b} = g(w) [f, K_{b}(·, w)]_{b} = g(w) f(w)

shows that T = M_{g} ∈ M(F_{b}). □

Let S(S_{b}) be the smallest sot-closed algebra in B(F_{b}) containing {S_{b}} and I, W(S_{b}) be the smallest wot-closed algebra in B(F_{b}) containing {S_{b}} and I, and W^{∗}(S_{b}) be the smallest weak^{∗}-closed algebra in B(F_{b}) containing {S_{b}} and I.

Theorem 8.5. We have W^{∗}(S_{b}) = W(S_{b}) = S(S_{b}) = {S_{b}}′′ = {S_{b}}′ = M(F_{b}).

Proof. It is easy to see that a commutant is sot-closed, and thus it is also wotclosed since it is a subspace; see [M, Theorem 4.2.7]. Thus M(F_{b}) is sot-closed and wot-closed by Theorem 8.4. The same result implies also W(S_{b}) = S(S_{b}).

It is clear that {S_{b}} ⊂ {S_{b}}′ since individual shifts commute with each other. Hence {S_{b}} ⊂ {S_{b}}′′ ⊂ {S_{b}}′ as well. Also W(S_{b}) ⊂ {S_{b}}′′ by the remarks in the previous paragraph. Moreover, weak^{∗}-convergence implies wot-convergence; see [M, p. 126]. Then W^{∗}(S_{b}) ⊂ W(S_{b}).

Finally, let M_{g} ∈ M(F_{b}). Then g ∈ F_{b} ∩ H^{∞} by the general remarks on multipliers in Section 5. By Definition 3.1 and Theorem 3.2, there is a sequence {p_{n}} of polynomials converging to g in ∥ · ∥_{F_{b}} and uniformly on compact subsets of 𝔹. Then

[p_{n}(S_{b})f, K_{b}(·, w)]_{b} = [p_{n} f, K_{b}(·, w)]_{b} = p_{n}(w) f(w)
⟶ g(w) f(w) = [g f, K_{b}(·, w)]_{b} = [M_{g}f, K_{b}(·, w)]_{b}

for any f ∈ F_{b} and w ∈ 𝔹. Thus p_{n}(S_{b}) → M_{g} in wot and M_{g} ∈ W(S_{b}). Since {p_{n}(S_{b})} is a sequence and F_{b} is separable by definition, also p_{n}(S_{b}) → M_{g} weak^{∗} and M(F_{b}) ⊂ W^{∗}(S_{b}) by [Con, Proposition 20.3 (a)]. □

9. Almost Normality

We now present several results on the commutators, hyponormality, and subnormality of the shift operators on F_{b} and F_{q}. Our unilateral shifts singly or as a tuple are never normal as are immediately seen by the results of Section 6. So we investigate the conditions under which they are hyponormal or essentially normal or subnormal.

Definition 9.1. We call a commuting N-tuple of operators T = (T_{1}, …, T_{N}) on a Hilbert space H tuple-hyponormal if T^{∗}T − T T^{∗} ≥ 0 using Notation 6.1. We call T tuple-essentially normal if T^{∗}T − T T^{∗} is compact. We call T tuple-subnormal if there is a Hilbert space H′ ⊃ H and on H′ a commuting N-tuple T′ = (T′_{1}, …, T′_{N}) of normal operators such that each T_{j} is the restriction of T′_{j} to H.

When N = 1, the concepts with the tuple prefix reduce to the usual concepts without it. The definition of tuple-subnormality is from [Ar3, Remark 3.10].

Theorem 9.2. The operator S_{b} is tuple-hyponormal if and only if the weight sequence satisfies (N + k) b_{k}^{2} ≥ (1 + k) b_{k−1} b_{k+1} for k = 1, 2, . . .. In particular, when N = 1, S_{b} is tuple-hyponormal if and only if {b_{k}} is a log-concave sequence.

So for N = 1, S_{q} is (tuple-)hyponormal if and only q ≥ −1; for N = 2, S_{q} is tuple-hyponormal if and only if q ≥ −5/2 or −4 ≤ q ≤ −3; for N ≥ 3, S_{q} is tuple-hyponormal if and only if q ≥ (1 − N − N^{2})/N or q ≤ −(1 + N).

Proof. By (25),

(S_{b}^{∗}S_{b} − S_{b}S_{b}^{∗})f = \frac{N}{b_{1}} f_{0} + ∑_{k=1}^{∞}(\frac{N + k}{1 + k} \frac{b_{k}}{b_{k+1}} − \frac{b_{k−1}}{b_{k}} ) f_{k}

and N/b_{1} > 0. The first statement follows from Theorem 3.5. The second statement follows from the definition of a concave sequence which is b_{k} ≥ (b_{k−1} + b_{k+1})/2.

Next, by (27) and (28), (S_{q}^{∗}S_{q} − S_{q}S_{q}^{∗})f is

= \begin{cases}\frac{N}{1 + N + q} f_{0} + ∑_{k=1}^{∞}[\frac{N + k}{1 + N + q + k} − \frac{k}{N + q + k} ]f_{k}, & if q > −(1+N), \\ (1−N −q)Nf_{0} + ∑_{k=1}^{∞}[\frac{(1−N −q+k) (N +k)}{(1 + k)^{2}} + \frac{N +q−k}{k} ]f_{k}, & if q ≤ −(1+N). \end{cases}

The coefficient of f_{0} is always ≥ 0.

For q > −(1 + N), the coefficient of f_{k} is ≥ 0 for all k = 1, 2, … if and only if q ≥ (1 − N − N^{2})/N, which also yields the inequalities q ≥ −1 for N = 1 and q ≥ −5/2 for N = 2.

For q ≤ −(1 + N), the coefficient of f_{k} with k ≥ 1 is nonnegative if and only if q (1−Nk +2k) ≥ −N +(N^{2} −3N +1) k +(1−N) k^{2}. The coefficient of q on the left shows that we need to look at the cases N = 1, N = 2, and N ≥ 3 separately. When N = 1, we get q ≥ −1, but we have to have q ≤ −(1+N) = −2; so there are no new q values. When N = 2, we get q ≥ −4, but we have to have q ≤ −(1 + N) = −3; so the new q values are −4 ≤ q ≤ −3. Finally, when N ≥ 3, the inequality for q gives an upper bound for q after a check for all k ≥ 1, but this upper bound is actually greater than −(1 + N); so in this case too tuple-hyponormality obtains for all q ≤ −(1 + N). □

The case N = 1 is outlined in [Sh, Lemma, p. 83].

The commutators of the individual S_{b_{j}} and their adjoints are written easily using the formulas in Section 6. However, unless more is known about the detailed values of {b_{k}}, these formulas do not yield any meaningful result. So for such commutators, we confine ourselves to the S_{q_{j}}. It turns out that these formulas are useful due to the powerful radial differential operators D_{s}^{t}.

Proposition 9.3. For i, j = 1, …, N, we have

S_{q_{i}}^{∗}S_{q_{j}}−S_{q_{j}}S_{q_{i}}^{∗} = \begin{cases}\frac{1}{1 + N + q} D_{1+q}^{−1} (δ_{ij} I − S_{q_{j}}S_{q_{i}}^{∗}), & if q > −(1+N), \\ (1−N −q) D_{−1+q}^{1}(D_{−N}^{−1})^{2}(δ_{ij}I −S_{q_{j}}S_{q_{i}}^{∗}) + K_{j}, & if q ≤ −(1+N), \end{cases}

where

K_{j}(z^{α}) = α_{j} (N + q)^{2} (D_{−N}^{−1})^{2} ((R′)^{−1})^{2} (z^{α}),

(R′)^{−1} is given in (32), and δ_{ij} is the Kronecker delta.

Proof. We show the details only for i = j; those for i ≠ j are similar and in fact simpler. Also it is enough to show the action of operators on z^{α} for |α| > 0.

For q > −(1 + N), using (26) and (19) and simplifying, we have

(S_{q_{j}}^{∗}S_{q_{j}} − S_{q_{j}}S_{q_{j}}^{∗}) z^{α} = \frac{N + q + |α| − α_{j}}{(1 + N + q + |α|) (N + q + |α|)} z^{α}.

and

\frac{1}{1 + N + q} D_{1+q}^{−1} (I − S_{q_{j}}S_{q_{j}}^{∗}) z^{α} = \frac{1}{1 + N + q + |α|}(1 − \frac{α_{j}}{N + q + |α|})z^{α}.

These two expressions are the same.

For q ≤ −(1 + N), we have instead

(S_{q_{j}}^{∗}S_{q_{j}} − S_{q_{j}}S_{q_{j}}^{∗}) z^{α} = (\frac{1 + α_{j}}{(1 + |α|)^{2}} (1 − N − q + |α|) − \frac{α_{j}}{|α|^{2}} (−N − q|α|))z^{α}

and

(1−N −q)D_{−1+q}^{1}(D_{−N}^{−1})^{2}(I −S_{q_{j}}S_{q_{j}}^{∗}) z^{α} = \frac{(1−N−q+|α|) (|α|^{2}+α_{j}(N +q−|α|))}{|α|^{2} (1 + |α|)^{2}} z^{α}.

The term K_{j}(z^{α}) exactly accounts for the difference between the right hand sides of the last two formulas. □

The case q = −N of Proposition 9.3 is in [Ar3, Proposition 5.3]. The following is in [AZ, Corollary 3.5] with a completely different proof; its q = −N case is in [Ar3, p. 192].

Corollary 9.4. Each S_{q_{j}} is hyponormal if and only if q ≥ −N.

Proof. By Proposition 4.4, all the radial differential operators encountered in Proposition 9.3 are positive. Consequently K_{j} ≥ 0 as well. So everything depends on the positivity of I − S_{q_{j}}S_{q_{j}}^{∗}. For α_{j} > 0, we have

(I − S_{q_{j}}S_{q_{j}}^{∗}) z^{α} = \begin{cases}\frac{N + q + |α| − α_{j}}{N + q + |α|} z^{α}, & if q > −(1 + N), \\ \frac{|α| (1 − α_{j}) + (N + q) α_{j}}{|α|^{2}} z^{α}, & if q ≤ −(1 + N). \end{cases}

The coefficient of all the z^{α} are positive if and only if q ≥ −N. If α_{j} = 0, then I − S_{q_{j}}S_{q_{j}}^{∗} = I anyway. □

The formulas developed above tell us also which operators are essentially normal.

Proposition 9.5. Every S_{q} is tuple-essentially normal and every S_{q_{j}} is essentially normal. Further, the commutator S_{q_{i}}^{∗}S_{q_{j}}−S_{q_{j}}S_{q_{i}}^{∗} ∈ C^{p}(F_{q}) if and only if p > N.

The case q = −N of the last claim is in [Ar3, Proposition 5.3]. This fact is also called p-essential normality and has recently been extended to more general situations; see, for example, [GW]. Note again that the condition p > N is indepenent of q.

Proof. Consider the formula for (S_{q}^{∗}S_{q} − S_{q}S_{q}^{∗})f obtained in the proof of Theorem 9.2. The coefficient of f_{k} there is ∼ 1/k and Theorem 3.5 gives the result about S_{q}.

Next consider the formula for S_{q_{j}}^{∗}S_{q_{j}} −S_{q_{j}}S_{q_{j}}^{∗} obtained in Proposition 9.3. There I − S_{q_{j}}^{∗}S_{q_{j}} is bounded by the work of Section 6. It is composed with a radial differential operator of total order −1 which is compact by Proposition 4.4. The only remaining operator K_{j} is also a radial differential operator of total order −3 (the coefficient α_{j} in K_{j}(z^{α}) indicates a derivative of order 1) and is also compact.

For the final claim, we first show that A_{ij} = δ_{ij} I − S_{q_{j}}S_{q_{i}}^{∗} is not compact. Suppose it is. If i = j, using (26) and summing on i = 1, …, N, we see that

∑_{i=1}^{N}(I − S_{q_{i}}S_{q_{i}}^{∗}) z^{α} = \frac{N^{2} + Nq + (N − 1)|α|}{N + q + |α|} z^{α}

is also compact. If i ≠ j, summing on i, j = 1, …, N, we see that

∑_{i,j=1}^{N}(A_{ij}^{∗}A_{ij}) z^{α} = \frac{|α|^{2}}{(N + q + |α|)^{2}} z^{α}

is also compact. But these diagonal operators are not compact by Theorem 3.5. Then the Schatten ideal membership of the commutator depends solely on the radial differential operator of total order −1 multiplying A_{ij} in Proposition 9.3. We finish as for the last claim in Proposition 4.4 obtaining the same condition. □

We turn to subnormality. If q ≥ −1, then F_{q} is a subspace of the Lebesgue space L_{q}^{2} with the same norm by Remark 4.7. Then S_{q} has the normal extension T = (T′_{1}, …, T′_{N}), where T′_{j} = M_{z_{j}} with (T′_{j})^{∗} = M_{\overline{z}_{j}}. Thus S_{q} is tuple-subnormal for q ≥ −1. [AZ, Theorem 6.1] contains the result that there are not any other tuple-subnormal S_{q} for q > −(1 + N); the case q = −N is given in [Ar3, p. 180]; see also [GHX, Theorem 3.2]. It turns out that there is not any other subnormal S_{q} for q ≤ −(1 + N). Our different proof depends on the following result which is important in its own right.

Theorem 9.6. Let C_{b} be an upper bound for the sequence {b_{k}^{1/k}} which is finite by (3). Then the geometric joint spectral radius of S_{b} is at most 1/\sqrt{C_{b}} and hence the Taylor joint spectrum of S_{b} lies in the closed ball of this radius.

See [Cu, Definition 2.1] for the Taylor joint spectrum σ_{T}. See [CŻ, p. 253] for the geometric joint spectral radius r_{G}.

Proof. We follow the proof of [GRS, Lemma 2.5], where a similar computation is done for S_{q} with q > −(1+N). In [CŻ, Theorem 1], r_{G} is shown to be independent of the particular joint spectrum, and a formula for it is given in [MS]. Using it and Proposition 3.4, we deduce that if we can find positive constants C_{α} such that ∥S_{b}^{α}z^{β}∥_{b} ≤ C_{α} ∥z^{β}∥_{b} for all multi-indices β, then

r_{G}(S_{b}) ≤ lim_{m→∞}( ∑_{|α|=m} \frac{m!}{α!} C_{α}^{2})^{1/2m}.

Note that |α + β| = |α| + |β|. Then

∥S_{b}^{α}z^{β}∥_{b}^{2} = ∥z^{α+β}∥_{b}^{2} = \frac{(α + β)!}{b_{|α|+|β|} (|α| + |β|)!} ≤ \frac{(α + β)!}{β!} \frac{C_{b}^{−|α|}}{(1 + |β|)_{|α|}} ∥z^{β}∥_{b}^{2},

and by (1) and (2),

∑_{|α|=m} \frac{m!}{α!} ∥S_{b}^{α}z^{β}∥_{b}^{2} = \frac{∥z^{β}∥_{b}^{2}}{C_{b}^{m}} \frac{(1)_{m}}{(1 + |β|)_{m}} ∑_{|α|=m} \frac{(α + β)!}{α! β!}

≤ C C_{b}^{−m} m^{−|β |} ∥z^{β}∥_{b}^{2} ∑_{|α|=m} \frac{(1 + β_{1})_{α_{1}}}{(1)_{α_{1}}} ⋯ \frac{(1 + β_{N})_{α_{N}}}{(1)_{α_{N}}}

≤ C C_{b}^{−m} m^{−|β |} ∥z^{β}∥_{b}^{2} m^{|β|} m^{N−1} = C C_{b}^{−m} m^{N−1} ∥z^{β}∥_{b}^{2}.

Taking the 2mth root of the coefficient and letting m → ∞ yield the desired result for S_{b}. □

Corollary 9.7. For any q ∈ ℝ, r_{G}(S_{q}) ≤ 1 and σ_{T} (S_{q}) ⊂ \overline{𝔹}.

Proof. We have C_{b(q)} = 1 for all q ∈ ℝ. □

Through other means, in [GHX, Theorem 4.5], it is shown that σ_{T} (S_{q}) = \overline{𝔹}. The same result also finds the Taylor joint essential spectrum of S_{q}, but we find it in a different way in Corollary 10.16 below.

Theorem 9.8. The operator S_{q} is tuple-subnormal if and only if q ≥ −1.

Proof. Only the only if part needs a proof. So let q < −1 and suppose S_{q} is tuple-subnormal. We use [At1, Theorem 5.2] which states that a tuple T is tuplesubnormal with σ_{T} (T) ⊂ \overline{𝔹} if and only if

h_{n}(T) := (I − J_{T^{∗}}(I))^{n} = ∑_{k=0}^{n}(−1)^{k}\binom{n}{k} ∑_{|α|=k}(T^{α})^{∗}T^{α} ≥ 0

for every n = 1, 2, . . .. The Taylor joint spectrum condition is provided by Corollary 9.7. We only need n = 1 to arrive at a contradiction. Then h_{1}(S_{q}) = I − S_{q}^{∗}S_{q} with Notation 6.1, and this is not positive if q < −1 by (31). □

Corollary 9.9. Among the F_{q}, the only spaces in which the norm of f ∈ F_{q} can be described by integration of |f|^{2} against a positive measure on ℂ^{N} are those with q ≥ −1.

Proof. This is of course [BB, Theorem 4.3], but our proof is very different and immediate by [GHX, Proposition 3.1], where the condition in the statement of the theorem is proved to be equivalent to tuple-subnormality of the shift operator. □

The limited case q > −(1 + N) of the following is also in [AZ, Corollary 3.5]; we give a completely different proof for all q ∈ ℝ.

Proposition 9.10. Each S_{q_{j}} is subnormal if and only if q ≥ −N.

Proof. For q < −N, S_{q_{j}} is not subnormal since it is not hyponormal by Corollary 9.4. For q ≥ −N, we use [Ag, Theorem 3.1] which states that an operator A is subnormal and ∥A∥ ≤ 1 if and only if A is an N-hypercontraction for every n = 1, 2, … meaning that

h_{n}(A) := (I − J_{A^{∗}}(I))^{n} = ∑_{k=0}^{n}(−1)^{k}\binom{n}{k}(A^{∗})^{k}A^{k} ≥ 0

for every n = 1, 2, . . .. For any n and multi-index α, we have

h_{n}(S_{q_{j}}) z^{α} = \frac{N + q + |α| − α_{j}}{1 + N + q + |α|} (1 − \frac{1 + α_{j}}{2 + N + q + |α|}(2 − \frac{2 + α_{j}}{3 + N + q + |α|} ·

· (3 − \frac{3 + α_{j}}{4 + N + q + |α|} ⋯ (n − 1 − \frac{n − 1 + α_{j}}{n + N + q + |α|}))))z^{α}.

It suffices to check the case q = −N. The largest possible number after the minus sign in the first parantheses is 1/2 and the smallest number after the minus sign in the second parantheses is 0. In such a case, the coefficient of z^{α} is 0. In all other cases, this coefficient is ≥ 0. Thus h_{n}(S_{q_{j}}) ≥ 0 for any n and j for q ≥ −N, and S_{q_{j}} is subnormal. □

Let’s record here that the hyponormality of an operator or a tuple T is deduced from the positivity of operators of the form I − T T^{∗} and its subnormality from the positivity of operators of the form I − T^{∗}T.

10. Toeplitz Algebras

We would like to investigate the C^{∗}-algebras generated by the shift operators on F_{q}. They are also called Toeplitz algebras because of their close connection to Toeplitz operators on weighted Bergman and Hardy spaces, those F_{q} with q ≥ −1. However, in the few publications dealing with q < −1, these algebras are defined only with shift operators. It is our aim now to put back the Toeplitz operators to the Toeplitz algebras. Our approach is classical, but brings out the properties of these algebras better.

For q > −1, the Dirichlet spaces F_{q} are the Bergman spaces A_{q}^{2} which are closed subspaces of the Lebesgue classes L_{q}^{2}. There are on them the usual Toeplitz operators T_{q,ϕ} : F_{q} → F_{q} whose symbols are measurable functions ϕ on 𝔹 defined by T_{q,ϕ} := P_{q}M_{ϕ}i, where i : F_{q} → L_{q}^{2} is the inclusion, M_{ϕ} : L_{q}^{2} → L_{q}^{2} denotes multiplication by ϕ, and P_{q} : L_{q}^{2} → F_{q} is the orthogonal projection. For ϕ(z) = z_{j}, we actually have T_{q,z_{j}} = S_{q_{j}}, j = 1, …, N.

For q ≤ −1, F_{q} need not be a subspace of L_{q}^{2} and the idea of projecting directly onto a subspace fails. In [AK3, Section 4], a more general family of Toeplitz operators acting on all F_{q} has been developed. The construction below works on any F_{q} with q ∈ ℝ and in fact yields often different Toeplitz operators even on the weighted Bergman spaces F_{q} for which q > −1.

Throughout this section, we use the integral norm ⫴·⫴_{q} and the associated inner product ⟦·, ·⟧_{q} on F_{q} introduced in Remark 4.7, because they give exact formulas and simplify our work.

Given q ∈ ℝ, fix s so that −q + 2s = 0 and fix also t = −q + s; then q + 2t = 0 and s + t = 0. (Actually, = 0 is used here for simplicity; > −1 works just as fine.) The generalized Toeplitz operator U_{q,ϕ} : F_{q} → F_{q} is defined by

U_{q,ϕ} := P_{s} M_{ϕ} I_{s}^{t} = P_{q/2} M_{ϕ} I_{q/2}^{−q/2},

where I_{s}^{t} : F_{q} → L_{q}^{2} is one of the imbeddings introduced in Theorem 4.6 and P_{s} : L_{q}^{2} → F_{q} is a generalized Bergman projection that is proved to be bounded in [K1, Theorem 1.2] for the selection of the parameters s, t. It must be noted that for q ≤ −1, we cannot use P_{q} in place of P_{q/2}. This form of a Toeplitz operator follows the classical form except that the usual inclusion i is replaced by a different imbedding I_{s}^{t}. In [AK3, Theorem 4.5], it is shown that

(39)  U_{q,ϕ} = (D_{s}^{t})^{−1} T_{0,ϕ} D_{s}^{t} = D_{0}^{q/2} T_{0,ϕ} D_{q/2}^{−q/2},

where T_{0,ϕ} is the usual Toeplitz operator on the unweighted Bergman space F_{0}. By Corollary 4.8, D_{s}^{t} moves us up to the Bergman space F_{0}, T_{0,ϕ} acts on this Bergman space, and then (D_{s}^{t})^{−1} moves us back to the Besov space F_{q}. From now on, we use T_{0,ϕ} only with first parameter q = 0. For q = 0 and only for it, U_{0,ϕ} = T_{0,ϕ} and U_{0,z_{j}} = T_{0,z_{j}} = S_{0_{j}}, because

U_{q,z_{j}} z^{α} = (D_{0}^{q/2} S_{0_{j}} D_{q/2}^{−q/2}) z^{α} = \frac{b_{|α|+1}(q/2)}{b_{|α|+1}(0)} \frac{b_{|α|}(0)}{b_{|α|}(q/2)} z^{α+e_{j}}

= \begin{cases}\frac{1 + N + q/2 + |α|}{1 + N + |α|} z^{α+e_{j}}, & if q/2 > −(1 + N), \\ \frac{(1 + |α|)^{2}}{(1 + N + |α|) (1 − (N + q/2) + |α|)} z^{α+e_{j}}, & if q/2 ≤ −(1 + N), \end{cases} (40)

by (19) and Definition 4.1. Note that the shift operator in the middle acts on the unweighted Bergman space F_{0}.

Let’s compute the adjoints. First for q > −1,

T_{q,ϕ}^{∗} = i^{∗}M_{ϕ}^{∗}P_{q}^{∗} = P_{q}M\overline{ϕ} i = T_{q,\overline{ϕ}}  and  T_{q,z_{j}}^{∗} = T_{q,\overline{z}_{j}} = S_{q_{j}}^{∗}

since i and P_{q} are adjoints of each other by [AK3, (3.15)]. More importantly,

(41)  U_{q,ϕ}^{∗} = (D_{q/2}^{−q/2})^{∗} T_{0,ϕ}^{∗} (D_{0}^{q/2})^{∗} = D_{0}^{q/2} T_{0,\overline{ϕ}} D_{q/2}^{−q/2} = U_{q,\overline{ϕ}}

since

(42)  (D_{s}^{t})^{∗} = (D_{s}^{t})^{−1} = D_{s+t}^{−t} : F_{q+2t} → F_{q}

by [AK3, (3.6)] using ⟦·, ·⟧_{q}. Further,

U_{q,z_{j}}^{∗} z^{α} = (D_{0}^{q/2} S_{0_{j}}^{∗} D_{q/2}^{−q/2}) z^{α} = \frac{α_{j}}{N + |α|} \frac{b_{|α|−1}(q/2)}{b_{|α|−1}(0)} \frac{b_{|α|}(0)}{b_{|α|}(q/2)} z^{α+e_{j}},

= \begin{cases}\frac{α_{j}}{N + q/2 + |α|} z^{α−e_{j}}, & if q/2 > −(1 + N), \\ \frac{(−(N + q/2) + |α|) α_{j}}{|α|^{2}} z^{α−e_{j}}, & if q/2 ≤ −(1 + N), \end{cases}

for α_{j} > 0. Of course U_{q,z_{j}}^{∗} z^{α} = 0 if α_{j} = 0 and so U_{q,z_{j}}^{∗} 1 = 0.

We state a few other properties of the U_{q,ϕ} obtained in [AK3, Theorem 4.5, Propositions 4.2 and 4.10] that we need. The first is just a repetition of (42), and the others follow from similar well-known results on Toeplitz operators on F_{0}.

Theorem 10.1. (i) The operators U_{q,ϕ} and T_{0,ϕ} are unitarily equivalent via D_{q/2}^{−q/2}. (ii) If ϕ ∈ L^{∞}, then ∥U_{q,ϕ}∥ ≤ ∥ϕ∥_{L^{∞}} using ⫴ · ⫴_{q} on F_{q}. (iii) For ϕ ∈ C(\overline{𝔹}), U_{q,ϕ} is compact if and only if ϕ restricts to 0 on ∂𝔹.

A straightforward computation shows that

U_{q,z_{j}}U_{q,z_{j}}^{∗} z^{α} = \frac{α_{j}}{N + |α|} z^{α} = S_{0_{j}}S_{0_{j}}^{∗}z^{α},
U_{q,z_{j}}^{∗}U_{q,z_{j}} z^{α} = \frac{1 + α_{j}}{1 + N + |α|} z^{α} = S_{0_{j}}^{∗}S_{0_{j}}z^{α},

It follows that by picking those formulas and results in Sections 6 and 9 for S_{0}, the shift operator on the unweighted Bergman space F_{0}, we obtain for free many properties for U_{q} = (U_{q,z_{1}}, …, U_{q,z_{N}}) for any q.

Our main purpose in the section is to investigate the so-called Toeplitz algebras that we now define.

Definition 10.2. Let q ∈ ℝ. We denote by T_{q} the C^{∗}-algebra generated by the shift operators S_{q_{1}}, …, S_{q_{N}} in B(F_{q}). That is, T_{q} is the operator-norm closure of polynomials in S_{q_{j}} and S_{q_{j}}^{∗}.

Our study of T_{q} requires the study of two other similar algebras.

Definition 10.3. We denote by U_{q} the C^{∗}-algebra generated by the Toeplitz operators U_{q,z_{1}}, …, U_{q,z_{N}}. We denote by V_{q} the C^{∗}-algebra of operators of the form U_{q,ϕ} + K, where ϕ ∈ C(\overline{𝔹}) and K ∈ K(F_{q}).

Proposition 10.4. The collection V_{q} is in fact a C^{∗}-algebra.

Proof. Throughout, ⫴ · ⫴_{q} is used. Let ϕ, ψ ∈ C(\overline{𝔹}). Then the semicommutator

U_{q,ϕ} U_{q,ψ} − U_{q,ϕψ} = D_{0}^{q/2} (T_{0,ϕ} T_{0,ψ} − T_{0,ϕψ}) D_{q/2}^{−q/2} = D_{0}^{q/2} K_{1} D_{q/2}^{−q/2} = K_{2},

where K_{1} ∈ K(F_{0}) by the proof of [Cob, Theorem 1], and K_{2} ∈ K(F_{q}). It follows that V_{q} is an algebra. It is clear that it is a ∗-algebra, that is, it is closed under taking adjoints. Aside, using the result on the semicommutator twice, the commutator U_{q,ϕ} U_{q,ψ} − U_{q,ψ} U_{q,ϕ} is also compact.

Suppose U_{q,ϕ_{n}} + K_{n} → X in norm. That is, D_{0}^{q/2} (T_{0,ϕ_{n}} + L_{n}) D_{q/2}^{−q/2} → X in

norm, where L_{n} = D_{q/2}^{−q/2} K_{n} D_{0}^{q/2} ∈ K(F_{0}). So T_{0,ϕ_{n}} + L_{n} → D_{q/2}^{−q/2} X D_{0}^{q/2} = Y. By [Cob, Theorem 1], Y = T_{0,ϕ_{0}} + K_{0} with K_{0} ∈ K(F_{0}). Hence X ∈ V_{q} and V_{q} is norm-closed.

Since V_{q} lies in B(F_{q}), we conclude that it is a C^{∗}-algebra. □

Proposition 10.5. The algebras T_{q} and U_{q} contain the identity operator.

Proof. The operator S_{q}^{∗}S_{q} lies in T_{q} and is invertible in B(F_{q}) by (28) and Theorem 3.5. By spectral permanence (see [Da, Corollary I.5.7]), this inverse lies in the C^{∗}- algebra generated by S_{q}^{∗}S_{q} which lies in T_{q}. Thus I ∈ T_{q}.

For U_{q}, we use U_{q}^{∗}U_{q} = S_{0}^{∗}S_{0} instead. □

Corollary 10.6. The double centralizer (multiplier) algebra of T_{q} is itself.

Proof. See [Bu, Section 2] for the definition and basic properties of the double centralizer algebra of a C^{∗}-algebra. By [Bu, Proposition 3.1], a C^{∗}-algebra is a closed two-sided ideal in its double centralizer algebra. □

Definition 10.7. A C^{∗}-algebra G ⊂ B(H) is called irreducible if there is no nontrivial (other than {0} and H) subspace of H which is invariant under all A ∈ G. Equivalently, G is irreducible if there is no nontrivial orthogonal projection P ∈ B(H) commuting with all A ∈ G.

Theorem 10.8. The algebras T_{q} and U_{q} are irreducible.

This result obviously is not new; see [JL, Corollary 13] for a different proof.

Proof. We follow [Cob, Lemma 1] which handles the cases q = −1 and q = 0. Suppose P is an orthogonal projection from F_{q} onto a closed subspace W commuting with all the elements in T_{q}. Let g = P 1 ∈ W. Then

P S_{q_{j}}(1) = P (z_{j}) = S_{q_{j}}P (1) = S_{q_{j}}(g) = z_{j}g (j = 1, …, N).

Repeating this with linear combinations of powers of S_{q_{j}}, we obtain that P f = gf for any f ∈ F_{q}, because linear combinations of powers of {z_{j}} is dense in F_{q} by Definition 3.1. Thus P = M_{g}. But σ(P) = σ(M_{g}) ⊂ {0, 1}. Since g ∈ H(𝔹), either g ≡ 0 or g ≡ 1. Equivalently, P = 0 or P = I, proving that nontrivial orthogonal projections on F_{q} commuting with T_{q} do not exist.

For U_{q}, we use Theorem 10.1 (i). If Q is an orthogonal projection from F_{q} onto a closed subspace W commuting with everything in U_{q}, then QU_{q,z_{j}} = U_{q,z_{j}}Q for each j = 1, …, N. Writing down the definition of U_{q,z_{j}} and using (42), we obtain P S_{q_{j}} = S_{q_{j}}P with P = D_{q/2}^{−q/2} Q D_{0}^{q/2} : F_{0} → D_{q/2}^{−q/2}(W) ⊂ F_{0} a projection. If

g = Q1, let h = P 1 = D_{q/2}^{−q/2}g ∈ H(𝔹). By the previous paragraph, P = M_{h}, and P = I or P = 0. Then also Q = I or Q = 0. □

Proposition 10.9. The algebras T_{q} and U_{q} include the ideal K(F_{q}).

Proof. The operators I − S_{q}S_{q}^{∗} and I − U_{q}U_{q}^{∗} = I − S_{0}S_{0}^{∗} are nonzero, compact by Corollary 6.14, and lie in T_{q} and U_{q}, respectively. Then Theorem 10.8 in conjunction with [Da, Corollary I.10.4] yields the desired result. □

Proposition 10.10. The operator U_{q,z_{j}} − S_{q_{j}} is compact and thus T_{q} = U_{q}.

Proof. By (40),

(U_{q,z_{j}} − S_{q_{j}})f = \begin{cases}∑_{k=0}^{∞} \frac{q/2}{1 + N + k} f_{k+1}, & if q/2 > −(1 + N), \\ ∑_{k=0}^{∞} \frac{N^{2} + (1 + N + k) q/2}{(1 + N + k) (1 − (N + q/2) + k)} f_{k+1}, & if q/2 ≤ −(1 + N). \end{cases}

By Theorem 3.5, U_{q,z_{j}} − S_{q_{j}} is compact. Thus by Proposition 10.9, U_{q,z_{j}} lies in T_{q} and S_{q_{j}} lies in U_{q}. □

Proposition 10.11. The algebras V_{q} and U_{q} are the same.

Proof. By Proposition 10.4, U_{q} ⊂ V_{q}. Conversely, let U_{q,ϕ} ∈ V_{q}, where ϕ ∈ C(\overline{𝔹}). There is a sequence \{ p_{n}(z, \overline{z})\} of polynomials converging to ϕ uniformly on \overline{𝔹} by the Stone-Weierstrass theorem. Since U_{q} is an algebra, U_{q,p_{n}} ∈ U_{q}. By Theorem 10.1 (ii), U_{q,p_{n}} → U_{q,ϕ} in norm and thus U_{q,ϕ} ∈ U_{q}. By Proposition 10.9, K(F_{q}) ⊂ U_{q}. Therefore V_{q} ⊂ U_{q} as well. □

Therefore

T_{q} = V_{q} = U_{q} (q ∈ ℝ).

Corollary 10.12. The commutator ideal of T_{q} is K(F_{q}).

Proof. The commutator ideal of T_{q} is by definition the closed ideal generated by all operators of the form AB − BA for A, B ∈ T_{q}. By the proof of Proposition 10.4, the commutators and hence the commutator ideal, which is nonempty, lie in the ideal K(F_{q}). But K(F_{q}) is a simple C^{∗}-algebra (see [Da, Corollary I.10.3]), that is, it contains no proper nonzero closed two-sided ideal. The result follows. □

Theorem 10.13. For every q ∈ ℝ,

0 ⟶ K(F_{q}) ⟶^{i_{q}} T_{q} ⟶^{γ_{q}} C(∂𝔹) ⟶ 0

is a short exact sequence of C^{∗}-algebras. Here i_{q} is the inclusion map and by definition γ_{q}(U_{q,ϕ} + K) := ϕ|_{∂𝔹} using the identification T_{q} = V_{q}, In particular, γ_{q}(S_{q_{j}}) = γ_{q}(U_{q,z_{j}}) = z_{j}|_{∂𝔹}. Also, there is a continuous section η_{q} : C(∂𝔹) → T_{q}.

Proof. Exact means that i_{q} is one-to-one, γ_{q} is onto, and ker(γ_{q}) = im(i_{q}). The first is by definition, and the third is an immediate consequence of Theorem 10.1 (iii). It is also clear that i_{q} is a ∗-algebra homomorphism.

Given Φ, Ψ ∈ C(∂𝔹), extend them to \overline{𝔹} using the Poisson integral and naming them ϕ, ψ. Then γ_{q}(U_{q,ϕ}) = Φ and γ_{q} is onto. The boundary values of ϕ ψ and of the Poisson integral χ of ΦΨ are both ΦΨ making the difference ϕ ψ − χ restrict to 0 on ∂𝔹. Then U_{q,χ} and U_{q,ϕψ} differ by an element of K(F_{q}) by Theorem 10.1 (iii). This shows that γ_{q} is an algebra homomorphism. It is also a ∗-homomorphism since the Poisson kernel is positive.

The action of γ_{q} on S_{q_{j}} is clear by Proposition 10.10.

Thus T_{q}/K(F_{q}) and C(∂𝔹) are isomorphic as C^{∗}-algebras through the map ι_{q}(U_{q,ϕ} + K(F_{q})) = ϕ|_{∂𝔹}. There is the induced quotient map θ_{q} : T_{q} → T_{q}/K(F_{q}) given by θ_{q}(A) := A + K(F_{q}).

Define η_{q} : C(∂𝔹) → T_{q} by η_{q}(Φ) := U_{q,ϕ}, which is continuous by Theorem 10.1 (ii). Then θ_{q} η_{q} is a ∗-algebra homomorphism and ι_{q} θ_{q} η_{q} = γ_{q} η_{q} = I. This η_{q} is the desired section. □

The use of the Poisson integral in the proof is somewhat arbitrary. Instead, we could use the invariant Poisson integral, which also has a positive kernel.

Remark 10.14. Theorem 10.13 covers the special cases of the Drury-Arveson space (q = −N, [Ar3, Theorem 5.7]), Hardy and Bergman spaces (q = −1 and q = 0, [Cob, Theorem 1]), and the Dirichlet space (q = −(1 + N), [Ca1, Theorem 8] for N = 1); after all, the definition of T_{q} is the same for all Hilbert spaces considered. Note that in the Hardy-space case, there is no need to extend continuous functions to all of \overline{𝔹}; most other sources work solely on ∂𝔹. In fact, the Hardy-space Toeplitz operators can be recovered in a limiting case as explained in [AK3, Remark 5.6]. Or, one can pass from the Hardy-space case (q = −1) to the Bergman-space case q = 0 as done in [Cob], and then pass to other q as done above. The coverage mentioned holds for all the other results in this article as well unless specifically stated.

Short exact sequences have been used to investigate a variety of C^{∗}-algebras of operators. See, for example, [J, Theorem 3.9] for those involving composition operators along with Toeplitz operators.

Corollary 10.15. The maximal ideal space of T_{q}/K(F_{q}) is ∂𝔹.

Proof. The maximal ideal space of C(K) is K. □

Corollary 10.16. The Taylor joint essential spectrum of S_{q} is ∂𝔹.

See [Cu, Definition 3.3] for the Taylor joint essential spectrum σ_{T_{e}}.

Proof. We use [Cu, Corollary 3.10] whose essential normality condition on S_{q} is provided by the second part of Proposition 9.5. Then σ_{T_{e}}(S_{q}) is the maximal ideal space of T_{q}/K(F_{q}), which is identified in Corollary 10.15. □

Proposition 10.17. We have ∥S_{q}S_{q}^{∗}∥_{e} = ∥S_{q}^{∗}S_{q}∥_{e} = ∥I∥ = 1.

Proof. These are immmediate by Corollary 6.14. □

Proposition 10.18. The identity representation of T_{q} is a boundary representation for the concrete operator space S_{q} = span{I, S_{q_{1}}, …, S_{q_{N}} } if and only if q < −1.

See [Ar3, p. 205] for the definition of boundary representation.

Proof. This has been done in [GHX] for q ≥ −(1 + N); see the discussion on p. 220. The method of [Ar3, Lemma 7.13] is applicable which takes care of the case q = −N. Various results in this section supply the conditions mentioned in [Ar2, Theorem 2.1.1]. Then the proof boils down to determining those values of q for which ∥S_{q}^{∗}S_{q}∥ > ∥S_{q}^{∗}S_{q}∥_{e} by [Ar2, Remark, p. 284]. By (30) and Proposition 10.17, this is the case exactly for q < −1, even when N = 1. □

Corollary 10.19. If q < −1, then the C^{∗}-envelope of S_{q} is the Toeplitz algebra T_{q}.

See [H, Definition 2.5] for the definition of C^{∗}-envelope.

Proof. Clearly, S_{q} generates T_{q} as a C^{∗}-algebra. By Proposition 10.18, the intersection of the kernels of all boundary representations of S_{q} is {0} since the identity representation has that kernel. By [Ar1, Theorem 2.2.3], the Shilov boundary of S_{q} is then {0}. By [H, Theorem 4.4, Proof], the C^{∗}-envelope is the quotient of T_{q} by the Shilov boundary and is T_{q}. □

Definition 10.20. An operator A ∈ B(H) is called Fredholm if both ker(A) and ker(A^{∗}) = im(A)^{⊥} are finite-dimensional, in which case the Fredholm index of A is defined as ind(A) = dim ker(A) − dim ker(A^{∗}). By the Atkinson theorem (see [RLL, Theorem 9.4.1]), equivalently, A is Fredholm if [A] is invertible in Q(H), and equivalently, A is Fredholm if there exists a B ∈ B(H) such that I − AB and I − BA are compact.

When N = 1, it is clear that every S_{b} is Fredholm with ind(S_{b}) = −1. But when N > 1, no single shift operator S_{b_{j}} is Fredholm, because S_{b_{j}}^{∗} annihilates the infinite-dimensional subspace of F_{b} consisting of those f whose Taylor series (12) have f_{α} = 0 whenever α_{j} > 0.

On the other hand, with Notation 6.1, S_{b}S_{b}^{∗} is Fredholm with ind(S_{b}S_{b}^{∗}) = 0 since it is self-adjoint and its kernel is one-dimensional, and S_{b}^{∗}S_{b} is Fredholm with ind(S_{b}^{∗}S_{b}) = 0 since it is invertible. The latter fact is equivalent to the Fredholmness and the invertibility of the tuple S_{b} by [Cu, Corollary 3.9]; see [Cu, Definition 3.1] for the definition.

Proposition 10.21. The operator I − S_{b}S_{b}^{∗} is Fredholm if and only if all except finitely many of the b_{k} are different. The operator I − S_{b}^{∗}S_{b} is Fredholm if and only if (1 + k) b_{k+1} ≠ (N + k) b_{k} for all except finitely many k, which reduces to the condition in the previous sentence for N = 1. The operator I − S_{q}S_{q}^{∗} is Fredholm if and only if q ≠ −N. The operator I − S_{q}^{∗}S_{q} is Fredholm if and only if q ≠ −1. In each case that an operator is Fredholm, its index is 0.

Proof. It is enough to have a look at the explicit forms of the operators on f ∈ F_{b} or f ∈ F_{q} in Section 6. □

Corollary 10.22. For ϕ ∈ C(\overline{𝔹}) and K ∈ K(F_{q}), U_{q,ϕ} + K is Fredholm if and only if ϕ does not vanish on ∂𝔹.

Proof. By spectral permanence, U_{q,ϕ} + K is a Fredholm operator if and only if [ U_{q,ϕ} + K] = [ U_{q,ϕ}] is invertible in T_{q}/K(F_{q}). By Theorem 10.13, this quotient is isomorphic to C(∂𝔹). Thus U_{q,ϕ} + K is Fredholm if and only if ϕ|_{∂𝔹} is invertible, that is, if and only if this restriction is never 0. □

Remark 10.23. If A and B are unitarily equivalent and A is Fredholm, then B is also Fredholm with ind(A) = ind(B). Thus the Fredholm properties of U_{q,ϕ} can in general be surmised from those of T_{0,ϕ} by Theorem 10.1 (i).

Corollary 10.24. If ϕ ∈ C(\overline{𝔹}), then σ_{e}(U_{q,ϕ}) = σ_{e}(T_{q,ϕ}) = ϕ(∂𝔹).

Proof. It suffices to do the proof for U_{q,ϕ} only. If λ ∉ ϕ(∂𝔹), then U_{q,ϕ} − λI is invertible in C(∂𝔹). Equivalently, U_{q,ϕ} − λI + K(F_{q}) = [ U_{q,ϕ} − λI] is invertible in T_{q}/K(H) by Theorem 10.13. By spectral permanence, equivalently, [ U_{q,ϕ} − λI] is invertible in Q(H). Hence λ ∉ σ_{e}(U_{q,ϕ}).

Conversely, let λ ∈ ϕ(∂𝔹). Without loss of generality, ϕ(1, 0, …, 0) = λ = 0. We need to show that U_{q,ϕ} is not Fredholm. If it were Fredholm, then T_{0,ϕ} would also be Fredholm by Remark 10.23. Then the construction in the proof of [Z, Theorem 6.5] with z replaced by z_{1} would yield a contradiction. □

Proposition 10.25. Suppose ϕ ∈ C(\overline{𝔹}) and U_{q,ϕ} is Fredholm. If N > 1, then ind(U_{q,ϕ}) = 0. If N = 1, then ind(U_{q,ϕ}) = −n, where n is the winding number of the closed curve ϕ(∂𝔻) about 0.

Proof. By Corollary 10.22, ϕ does not vanish on ∂𝔹. For N > 1, 𝔹 is strongly pseudoconvex and has simply connected boundary. Then [V, Theorem 1.4] shows that ind(T_{0,ϕ}) = 0. By Remark 10.23, also ind(U_{q,ϕ}) = 0. For N = 1, by [V, Theorem 1.3], ind(T_{0,ϕ}) = −n. Then by Remark 10.23, also ind(U_{q,ϕ}) = −n. □

Corollary 10.26. The short exact sequence in Theorem 10.13 is not split exact if N = 1.

Proof. Split exact means that there is a ∗-algebra homomorphism κ_{q} : C(∂𝔻) → T_{q} such that γ_{q} κ_{q} = I. Suppose that is the case. Let Φ ∈ C(∂𝔻) be given by Φ(ζ) = ζ = z|_{∂𝔻} and let N_{q} = κ_{q}(Φ) ∈ T_{q}. By assumption, γ_{q}(N_{q}) = γ_{q}(U_{q,z}) = ζ. On the other hand, since κ_{q}(\overline{Φ}) = N_{q}^{∗} and \overline{Φ} commutes with Φ, N_{q} is normal. Then N_{q} − U_{q,z} = K_{q} ∈ ker(γ_{q}) = im(i_{q}) = K(F_{q}). Also U_{q,z} = D_{0}^{q/2} T_{0,z} D_{q/2}^{−q/2} is Fredholm since T_{0,z} = S_{0} is. But then N_{q} is also Fredholm and these imply the impossibility 0 = ind(N_{q}) = ind(U_{q,z}) = −1 by Proposition 10.25. □

We finish this section with a different kind of result.

Theorem 10.27. The algebra T_{q} is dense in B(F_{q}) in the strong operator topology.

Proof. Let A ∈ B(F_{q}). Consider Corollary 4.8 with q + 2t = 0 and s = q/2; then t = −q/2 and u = 0. Then as in the discussion at the beginning of this section, D_{q/2}^{−q/2} A D_{0}^{q/2} ∈ B(F_{0}). By [E, Corollary 3], the collection \{ T_{0,ϕ} : ϕ ∈ C(\overline{𝔹}) \} is dense in B(F_{0}) in the same topology. So there is a sequence \{ϕ_{n}\} with ϕ_{n} ∈ C(\overline{𝔹}) such that ⫴T_{0,ϕ_{n}}g−D_{q/2}^{−q/2} A D_{0}^{q/2}g⫴_{0} → 0 as n → ∞ for all g ∈ F_{0}. Put f = D_{0}^{q/2}g;

then f ∈ F_{q} and is arbitrary. By (39), we have ⫴D_{q/2}^{−q/2} U_{q,ϕ_{n}}f − D_{q/2}^{−q/2} Af⫴_{0} → 0 for all f ∈ F_{q}. This is the same as ⫴U_{q,ϕ_{n}}f − Af⫴_{q} → 0 as n → ∞ for all f ∈ F_{q}. We are done by \{ U_{q,ϕ} : ϕ ∈ C(\overline{𝔹}) \} ⊂ V_{q} = T_{q}. □

11. Ext Classes

This section is devoted to proving that all the short exact sequences indexed by q ∈ ℝ in Theorem 10.13 are in the same Ext class in two different ways. We notice that the maps that realize these are the radial differential operators that have been so useful already. In the process, we compute the Busby maps by interpreting the short exact sequences as extensions of C(∂𝔹). We also identify the K-theory groups K_{0}(T_{q}) and K_{1}(T_{q}); see [W] for their definitions and properties.

The bases of Proposition 3.4, when written for the F_{q} using (18), take the form

O_{α}^{q}(z) = \begin{cases}\begin{matrix}√ & \overline{(1 + N + q)_{|α|}} \\ √ & \overline{α!} \end{matrix} z^{α}, & if q > −(1 + N), \\ \frac{|α|!}{\sqrt{(1 − (N + q))_{|α|} α!}} z^{α}, & if q ≤ −(1 + N), \end{cases}

and are orthonormal with respect to ∥ · ∥_{q}. Given q_{1} ≠ q_{2}, define Q : F_{q_{1}} → F_{q_{2}} by Q(O_{α}^{q_{1}}) := O_{α}^{q_{2}}. This is a unitary operator with respect to [·, ·]_{q} by definition, and

Qz^{α} = \begin{cases}\sqrt{\frac{(1 + N + q_{2})_{|α|}}{(1 + N + q_{1})_{|α|}}} z^{α}, & if q > −(1 + N), \\ \sqrt{\frac{(1 − (N + q_{1}))_{|α|}}{(1 − (N + q_{2}))_{|α|}}} z^{α}, & if q ≤ −(1 + N). \end{cases}

The coefficient of z^{α} depends only on |α|; calling it g_{|α|}, we see that if f ∈ F_{q_{1}},

then Qf = ∑_{k=0}^{∞} g_{k} f_{k}, where g_{k} ∼ k^{(q_{2}−q_{1})/2}. Thus Q is a radial differential operator

of order (q_{2} − q_{1})/2 much like those in Section 4.

However, as in Section 10, we continue to use ⫴ · ⫴_{q} and ⟦·, ·⟧_{q}, and the correct radial differential operator to use with them is not Q but D_{q_{1}/2}^{(q_{2}−q_{1})/2} : F_{q_{1}} → F_{q_{2}} which is bounded and onto by Corollary 4.8. By (42), it is also unitary with respect to ⟦·, ·⟧_{q}, and

(43)  (D_{q_{1}/2}^{(q_{2}−q_{1})/2})^{∗} = (D_{q_{1}/2}^{(q_{2}−q_{1})/2})^{−1} = D_{q_{2}/2}^{(q_{1}−q_{2})/2} : F_{q_{2}} → F_{q_{1}}.

It can be checked that D_{q_{1}/2}^{(q_{2}−q_{1})/2} takes an orthonormal basis to another with respect to ⫴ · ⫴_{q}.

Proposition 11.1. For every q_{1}, q_{2} ∈ ℝ, there is a unital ∗-algebra isomorphism Ω between the C^{∗}-algebras K(F_{q_{1}}) and K(F_{q_{2}}), between B(F_{q_{1}}) and B(F_{q_{2}}), and between T_{q_{1}} and T_{q_{2}}, which induces a ∗-algebra isomorphism between Q(F_{q_{1}}) and Q(F_{q_{2}}).

Proof. If A is an operator in one of the C^{∗}-algebras with parameter q_{1}, define Ω(A) := D_{q_{1}/2}^{(q_{2}−q_{1})/2} A D_{q_{2}/2}^{(q_{1}−q_{2})/2}. Then Ω(A) lies in the corresponding C^{∗}-algebra

with parameter q_{2}. This Ω is invertible with Ω^{−1}(B) = D_{q_{2}/2}^{(q_{1}−q_{2})/2} B D_{q_{1}/2}^{(q_{2}−q_{1})/2} by (43), which also shows that Ω is an algebra isomorphism. The fact that Ω respects the ∗-operation follows from the positivity of the D_{s}^{t} stated in Proposition 4.4. Clearly, Ω(I) = I. □

The map Ω is often written Ad_{D_{q_{1}/2}^{(q_{2}−q_{1})/2}} in the literature.

Corollary 11.2. For every q ∈ ℝ, two groups arising in K-theory are K_{0}(T_{q}) = ℤ and K_{1}(T_{q}) = {0}.

Proof. The reference [Ca2, p. 178] gives these groups for q = −1; Remark 10.14 is relevant here. But these groups are invariant under homotopy equivalence between unital C^{∗}-algebras, hence under ∗-algebra isomorphisms; see [RLL, Propositions 3.2.6 and 8.2.2]. Proposition 11.1 finishes the proof. □

For only integer q ≥ −1, this result is in [At2, Proposition 3.1].

A short exact sequence as in Theorem 10.13, or the triple (i_{q}, T_{q}, γ_{q}), is often called an extension of C(∂𝔹) by K(F_{q}). There are two notions of equivalence between extensions in [Bl, 15.4] that we use. The first is the following.

Definition 11.3. We say two extensions of a C^{∗}-algebra G by H_{1} and H_{2} with middle terms E_{1} and E_{2} are weakly isomorphic if there are ∗-algebra isomorphisms ξ_{1} : G → G, ξ_{2} : E_{1} → E_{2}, ξ_{3} : H_{1} → H_{2} such that these maps and the maps of the extensions commute.

Theorem 11.4. For every q_{1}, q_{2} ∈ ℝ, extensions of C(∂𝔹) by K(F_{q_{1}}) and K(F_{q_{2}}) with middle terms T_{q_{1}} and T_{q_{2}} are weakly isomorphic.

Proof. Define ξ_{1} = I and ξ_{2} = ξ_{3} = Ω of Proposition 11.1. Then we have the following diagram.

0 ⟶ K(F_{q_{1}}) ⟶^{i_{q_{1}}} T_{q_{1}} ⟶^{γ_{q_{1}}} C(∂𝔹) ⟶ 0
↓^{Ω}  ↓^{Ω}  ↓^{I}
0 ⟶ K(F_{q_{2}}) ⟶^{i_{q_{2}}} T_{q_{2}} ⟶^{γ_{q_{2}}} C(∂𝔹) ⟶ 0

But by (39) and (20),

\begin{aligned}U_{q_{2},ϕ} & = D_{0}^{q_{2}/2} T_{0,ϕ} D_{q_{2}/2}^{−q_{2}/2} = D_{q_{1}/2}^{(q_{2}−q_{1})/2} D_{0}^{q_{1}/2} T_{0,ϕ} D_{q_{1}/2}^{−q_{1}/2} D_{q_{2}/2}^{(q_{1}−q_{2})/2} \\ & = D_{q_{1}/2}^{(q_{2}−q_{1})/2} U_{q_{1},ϕ} D_{q_{2}/2}^{(q_{1}−q_{2})/2} = Ω(U_{q_{1},ϕ}). \end{aligned} ; (44)

The maps on the left square commute trivially by Proposition 11.1. For the commutativity on the right square, let U_{q_{1},ϕ} + K_{1} ∈ T_{q_{1}} with K_{1} ∈ K(F_{q_{1}}). Then γ_{q_{2}}(Ω(U_{q_{1},ϕ} + K_{1})) = γ_{q_{2}}(U_{q_{2},ϕ} + K_{2}) = ϕ|_{∂𝔹}, where K_{2} ∈ K(F_{q_{2}}). But also I(γ_{q_{1}}(U_{q_{1},ϕ} + K_{1})) = ϕ|_{∂𝔹}. □

There is also the standard short exact sequence

0 ⟶ K(F_{q}) ⟶^{i_{q}} B(F_{q}) ⟶^{π_{q}} Q(F_{q}) ⟶ 0

of C^{∗}-algebras, where the inclusion map i_{q} and the quotient map π_{q} are ∗-algebra homomorphisms. If A ∈ B(F_{q}), then π_{q}(A) = [A] and π_{q}(A)^{∗} = π_{q}(A^{∗}). We also use a version of the standard short exact sequence in which operators act between different Hilbert spaces, which is

0 ⟶ K(F_{q_{1}}→F_{q_{2}}) ⟶^{i_{q_{1}q_{2}}} B(F_{q_{1}}→F_{q_{2}}) ⟶^{π_{q_{1}q_{2}}} Q(F_{q_{1}}→F_{q_{2}}) ⟶ 0.

Associated to each extension of a C^{∗}-algebra G by K(F_{q}) with middle term E, there are ∗-algebra homomorphisms ρ_{q} and τ_{q} as shown making the diagram commute, and τ_{q} is called the Busby map (see [Bu]):

0 ⟶ K(F_{q}) ⟶ E ⟶ G ⟶ 0
∥  ↓^{ρ_{q}}  ↓^{τ_{q}}
0 ⟶ K(F_{q}) ⟶^{i_{q}} B(F_{q}) ⟶^{π_{q}} Q(F_{q}) ⟶ 0

The Busby map is an invariant of the extension, that is, each determines the other; see [Bl, 15.3].

Given Φ ∈ C(∂𝔹), let ϕ be its Poisson integral. Working out the details of [M, Example 3.1.2] with G = C(∂𝔹), E = T_{q}, and i_{q} and γ_{q} as in Theorem 10.13, we compute that ρ_{q} = i_{q} and that the Busby map of the extension in Theorem 10.13 has the precise form

(45)  τ_{q}(Φ) = π_{q}(U_{q,ϕ}) = [ U_{q,ϕ}],

the equivalence class of the generalized Toeplitz operator U_{q,ϕ} in Q(F_{q}). This τ_{q} is one-to-one, because if τ_{q}(Φ) = [ 0 ] = K(F_{q}), then U_{q,ϕ} is compact which happens precisely when Φ = 0 by Theorem 10.1 (iii). This has a consequence. An ideal of a C^{∗}-algebra is called essential if it intersects every other nonzero ideal nontrivially. By [W, Proposition 3.2.15], we conclude the following.

Corollary 11.5. The ideal K(F_{q}) is essential in T_{q}.

In this connection, let’s recall that the maximal unital C^{∗}-algebra in which K(F_{q}) is essential is its double centralizer (multiplier) algebra B(F_{q}); see [Bl, 12.1].

The second equivalence notion in [Bl, 15.4] that we use is the following.

Definition 11.6. We say two extensions of a C^{∗}-algebra G by K(F_{q_{1}}) and K(F_{q_{2}}) with Busby maps τ_{q_{1}} and τ_{q_{2}} are strongly unitarily equivalent if there is a unitary operator V : F_{q_{1}} → F_{q_{2}} such that for all A ∈ G, we have

(46)  τ_{q_{2}}(A) = π_{q_{1}q_{2}}(V) τ_{q_{1}}(A) π_{q_{1}q_{2}}(V)^{∗}  (A ∈ G)

in Q(F_{q_{2}}). We write Ext(G) for the equivalence classes of extensions of G under this notion.

Theorem 11.7. For every q_{1}, q_{2} ∈ ℝ, extensions of C(∂𝔹) by K(F_{q_{1}}) and K(F_{q_{2}}) are strongly unitarily equivalent. In other words, all these extensions represent the same class in Ext(C(∂𝔹)).

Proof. Let Φ ∈ C(∂𝔹) and ϕ be its Poisson integral as usual. By (44), we have U_{q_{2},ϕ} = D_{q_{1}/2}^{(q_{2}−q_{1})/2} U_{q_{1},ϕ} D_{q_{2}/2}^{(q_{1}−q_{2})/2}. Define V : F_{1} → F_{2} as V = D_{q_{1}/2}^{(q_{2}−q_{1})/2}. It is unitary by (43). Apply π_{q_{2}} to the equality between the Toeplitz operators, use the fact that it is a homomorphism, and recall (45). Then

τ_{q_{2}}(Φ) = π_{q_{2}}(U_{q_{2},ϕ}) = π_{q_{2}}(V U_{q_{1},ϕ} V^{∗})
= π_{q_{1}q_{2}}(V) π_{q_{1}}(U_{q_{1},ϕ}) π_{q_{2}q_{1}}(V^{∗}) = π_{q_{1}q_{2}}(V) τ_{q_{1}}(Φ) π_{q_{1}q_{2}}(V)^{∗}.

We have achieved more than we have been aiming. The equality (46) holds not only at the equivalence class level, but also at the operator level with the V chosen. Thus a radial differential operator plays a key role once again. □

A similar theorem involving C^{∗}-algebras generated by Toeplitz and composition operators together, but only with q = 0 and q = −1, is in [J, Theorem 4.1].

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Department of Mathematics, Bi̇lkent University, 06800 Ankara, Turkey

   E-mail address: kaptan@fen.bilkent.edu.tr
   URL: http://www.fen.bilkent.edu.tr/∼kaptan/