UNITARY WEIGHTED COMPOSITION OPERATORS ON BERGMAN-BESOV AND HARDY HILBERT SPACES ON THE

BALL

H. TURGAY KAPTANOĞLU

Abstract. On weighted Bergman and Hardy Hilbert spaces on the unit ball of ℂ^{N}, we consider unitary weighted compositon operators T_{ψ} by the automorphisms ψ of the ball. Assuming that the homogeneous expansion of an f in one of these spaces contains only terms with total degree even (odd) and of T_{ψ}f contains only terms with total degree odd (even), we prove that f = 0. We also find related operators on the remaining Bergman-Besov Hilbert spaces including Drury-Arveson and Dirichlet spaces for which the same result holds. Our results generalize those obtained in [M] to the ball and to a wider family of function spaces.

1. Introduction

In a recent paper [M], the author proves that specific unitary weighted composition operators by the automorphisms of the unit disc on three (Bergman, Hardy, and Dirichlet) Hilbert spaces of holomorphic functions have the property that if a function in one of these spaces and its image under the corresponding operator have different parity, then it is the zero function.

Our objective in this paper is to extend this result to a wider classes of Hilbert spaces of holomorphic functions and to the case of the unit ball of ℂ^{N}. Moving up to arbitrary-dimensional balls where mappings of several complex variables are used complicate matters considerably. The geometry of Möbius transformations in the ball is more complicated and simple derivatives in the disc need to be replaced by complex Jacobians whose fractional powers are used in the generalizations of the operators of interest to the weighted spaces. Further, Besov Hilbert spaces such as the Dirichlet space have to be handled differently, because the derivatives used in the integral norms of such spaces simply are not compatible with the natural unitary weighted composition operators on them.

To present our result, we now introduce the necessary definitions and notation. Let 𝔹 be the open unit ball in ℂ^{N} with respect to the usual Hermitian inner product ⟨z, w⟩ = z_{1}\overline{w}_{1} + ⋯ + z_{N} \overline{w}_{N}, and the associated norm |z| = \sqrt{⟨z, z⟩}. When N = 1, the unit ball is the unit disc 𝔻 in the complex plane.

Date: 7 June 2023.

2020 Mathematics Subject Classification. Primary 47B33; Secondary 47B32, 47B38, 32A36, 32A35, 32A37, 46E20, 46E22.

Key words and phrases. Weighted composition operator, unitary operator, radial derivative, Bergman-Besov space, Hardy space, Dirichlet space, Drury-Arveson space.

Definition 1.1. For q ∈ ℝ and z, w ∈ 𝔹, the Bergman-Besov kernels are

K_{q}(z, w) :=\begin{cases}\frac{1}{(1−⟨z, w⟩)^{1+N+q}} =∑_{k=0}^{∞} \frac{(1+N +q)_{k}}{k!} ⟨z, w⟩^{k}, & q > −(1+N), \\ _{2}F_{1}(1, 1; 1−(N +q); ⟨z, w⟩) =∑_{k=0}^{∞} \frac{k! ⟨z, w⟩^{k}}{(1−(N +q))_{k}} , & q ≤ −(1+N), \end{cases}

where _{2}F_{1} ∈ H(𝔻) is the Gauss hypergeometric function and (a)_{b} is the Pochhammer symbol.

Definition 1.2. For q ∈ ℝ, the Bergman-Besov Hilbert space D_{q} is the reproducing kernel Hilbert space on 𝔹 generated by the kernel K_{q} endowed with the inner product and norm induced by K_{q}.

The kernels K_{q} are sesquiholomorphic on 𝔹^{2} and hence the functions in the D_{q} are holomorphic on 𝔹. In particular, D_{q} is the standard weighted Bergman space A_{q}^{2} for q > −1, the Hardy space H^{2} for q = −1, the Drury-Arveson space A for q = −N, and the Dirichlet space D for q = −(1 + N), that is, D_{−(1+N)} = D.

Let H(𝔹) be the space of all holomorphic functions on 𝔹. Every f ∈ H(𝔹) and thus every f ∈ D_{q} has homogeneous and Taylor expansions

(1)  f(z) = ∑_{k=0}^{∞} f_{k}(z) = ∑_{|α|=0}^{∞} f_{α}z^{α}  (z ∈ 𝔹)

converging absolutely and uniformly on compact subsets of 𝔹, where f_{k} is a homogeneous polynomial of degree k in z_{1}, …, z_{N}, α is a multi-index, and k = |α|. We use the expression f ∈ H(𝔹) has even parity (respectively, odd parity) to mean that the homogeneous expansion of f as in (1) has f_{k} with only even k (respectively, only odd k).

Denote by M the group of all one-to-one onto holomorphic maps (automorphisms) of 𝔹. Let Jψ be the complex Jacobian of ψ ∈ M. For ψ ∈ M, also ψ^{−1} ∈ M and Jψ ≠ 0 on 𝔹.

Definition 1.3. For q ≥ −(1 + N), ψ ∈ M and f ∈ D_{q}, define the operator T_{ψ}^{q} : D_{q} → D_{q} by

T_{ψ}^{q}f(z) := f(ψ(z))(Jψ(z))^{1+ \frac{q}{1+N}}

using an appropriate, say the principal, branch of the logarithm for the fractional power of Jψ(z).

So T_{ψ}^{q} is the product

T_{ψ}^{q} = M_{θ_{ψ}^{q}}C_{ψ},

where C_{ψ} is the composition operator given by C_{ψ}f = f ∘ ψ and M_{θ_{ψ}^{q}} is the multiplication operator by

(2)  θ_{ψ}^{q}(z) = (Jψ(z))^{1+ \frac{q}{1+N}} .

When q = −(1 + N), T_{ψ}^{−(1+N)} reduces simply to C_{ψ} on the Dirichlet space. When

q = 0, θ_{ψ}^{0} = Jψ for the unweighted Bergman space. When q = −1, θ_{ψ}^{−1} = (Jψ)^{\frac{N}{1+N}}

for the Hardy space. When q = −N, θ_{ψ}^{−N} = (Jψ)^{\frac{1}{1+N}} for the Drury-Arveson space.

In [BB, Theorem 1.10], it is proved that T_{ψ}^{q} is a unitary operator for q > −(1+N) with respect to the standard inner product that the kernel K_{q} induces on D_{q} given in (5) below. By [Z, Section 6.4], T_{ψ}^{−(1+N)} is unitary on the space D_{0} = D/ℂ with respect to the slightly different inner product (6).

Our main result is the following.

Theorem 1.4. Let q ≥ −1 and ψ ∈ M. Suppose the homogeneous expansions (1) of an f ∈ D_{q} and of T_{ψ}^{q}f are of different parity, that is, one contains terms only with k even and the other only with k odd. Then f = 0. There are also operators on D_{q} for q < −1 for which the same conclusion is true.

We prove Theorem 1.4 in Section 3. In the next Section 2, we provide further details and properties on notation, the spaces, and the automorphisms.

2. Preliminaries

In multi-index notation, α = (α_{1}, …, α_{N}) is an N-tuple of nonnegative integers, |α| = α_{1} + ⋯ + α_{N}, α! = α_{1}! ⋯ α_{N}!, 0^{0} = 1, and z^{α} = z_{1}^{α_{1}} ⋯ z_{N}^{α_{N}}. An overbar \overline{( )} indicates complex conjugate for numbers and functions and closure for sets. The boundary of 𝔹 is the unit sphere 𝕊.

The Pochhammer symbol (a)_{b} is defined by

(a)_{b} := \frac{Γ(a + b)}{Γ(a)}

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

(3) \frac{Γ(c + a)}{Γ(c + b)} ∼ c^{a−b},  \frac{(a)_{c}}{(b)_{c}} ∼ c^{a−b},  \frac{(c)_{a}}{(c)_{b}q} ∼ c^{a−b}  (Re c → ∞),

where A ∼ B means that |A/B| is bounded above and below by two strictly positive constants, that is, A = O(B) and B = O(A) for all A, B of interest.

The Gauss hypergeometric function _{2}F_{1} ∈ H(𝔻) is defined by

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

2.1. Spaces. A function K(z, w) is called the reproducing kernel of a Hilbert space H of functions defined on 𝔹 and with inner product ⟨·, ·⟩_{H} if K(·, w) ∈ H for each w ∈ 𝔹 and

u(z) = ⟨u(·), K(z, ·)⟩_{H} (u ∈ H, z ∈ 𝔹).

There is a one-to-one correspondence between reproducing kernel Hilbert spaces and positive definite kernels.

Let c_{k}(q) be the coefficient of ⟨z, w⟩^{k} in the series for K_{q}(z, w). Then c_{0}(q) = 1, c_{k}(q) > 0 for al k, and by (3),

(4)  c_{k}(q) ∼ k^{N+q}  (k → ∞),

for every q. This explains the choice of the parameters of the hypergeometric function in K_{q} for q < −(1 + N). The positive definiteness of ⟨z, w⟩ and the positivity of the c_{k}(q) yield that the K_{q} are positive definite and thus reproducing kernels. The kernels K_{q} for q < −(1+N) appear in the literature first in [BB, p. 13]. The kernels K_{q} for q > −(1+N) can also be written as _{2}F_{1}(1, 1+(N +q); 1; ⟨z, w⟩). For q < −(1+N), the functions in D_{q} are bounded on 𝔹 while the other D_{q} contain unbounded functions.

All Bergman-Besov kernels can be written of the form

K_{q}(z, w) = ∑_{k=0}^{∞} c_{k}(q)⟨z, w⟩^{k} = ∑_{k=0}^{∞} c_{k}(q) ∑_{|α|=k} \frac{|α|!}{α!}z^{α}\overline{w}^{α}.

Then by the theory of reproducing kernel Hilbert spaces and (4), the space D_{q} consists of all f ∈ H(𝔹) with Taylor expansions as in (1) for which

∥f∥_{D_{q}}^{2} := ∑_{|α|=0}^{∞} |f_{α}|^{2}∥z^{α}∥_{D_{q}}^{2} := ∑_{|α|=0}^{∞} |f_{α}|^{2} \frac{1}{c_{|α|}(q)} \frac{α!}{|α|!} ∼ ∑_{|α|=1}^{∞} |f_{α}|^{2} \frac{1}{|α|^{N+q}} \frac{α!}{|α|!} < ∞

equipped with the inner product

(5)  ⟨f, g⟩_{D_{q}} := ∑_{|α|=0}^{∞} \frac{1}{c_{|α|}(q)} \frac{α!}{|α|!}f_{α}\overline{g}_{α}.

The case of the Drury-Arveson space is especially simple, because then q = −N and c_{k}(−N) = 1 for all k = 1, 2, . . .. For q > −(1 + N), it is with respect to the inner product in (5) that the operators T_{ψ}^{q} are unitary.

Notice that the reproducing kernel of the Dirichlet space is

K_{−(1+N)}(z, w) = \frac{1}{⟨z, w⟩} log \frac{1}{1 − ⟨z, w⟩} = ∑_{k=0}^{∞} \frac{1}{1 + k} ⟨z, w⟩^{k}

and this gives

⟨f, g⟩_{D} = ∑_{|α|=0}^{∞}(1 + |α|)\frac{α!}{|α|!}f_{α}\overline{g}_{α}

with which ∥1∥_{D} = 1. The inner product with respect to which the operator T_{ψ}^{−(1+N)} = C_{ψ} is unitary on D_{0} = D/ℂ is

(6)  ⟨f, g⟩_{D_{0}} = ∑_{|α|=1}^{∞} |α|\frac{α!}{|α|!}f_{α}\overline{g}_{α}

with which ∥1∥_{D_{0}} = 0.

For s, t ∈ ℝ, we define the radial fractional differential operator D_{s}^{t} on H(𝔹) by

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

We have d_{0}(s, t) = 1 so that D_{s}^{t}(1) = 1, d_{k}(s, t) > 0 for any k, and by (4),

d_{k}(s, t) ∼ k^{t} (k → ∞),

for any s, t. So D_{s}^{t} is a continuous operator on H(𝔹) and is of order t. In particular, D_{s}^{t}z^{α} = d_{|α|}(s, t)z^{α} for any multi-index α. More importantly,

(7)  D_{s}^{0} = I,  D_{s+t}^{u}D_{s}^{t} = D_{s}^{t+u},  and  (D_{s}^{t})^{−1} = D_{s+t}^{−t}

for s, t, u ∈ ℝ, where the inverse is two-sided. Here and in any other context, I is the identity operator, Any D_{s}^{t} maps H(𝔹) onto itself continuously.

The d_{k}(s, t) are chosen the way they are in order to have

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

where differentiation is performed on the holomorphic variable z. More interestingly, by [AK, Proposition 3.2],

(8)  D_{s}^{t}(D_{q}) = D_{q+2t}

is an isomorphism of Hilbert spaces for any s, t and an isometry when the norms are chosen suitably.

The spaces D_{q} have also equivalent inner products and norms that are integrals of functions or their sufficiently high-order derivatives. For fixed q ∈ ℝ, let s, t ∈ ℝ be such that q + 2t > −1. By [AK, Definition 3.1c], a family of norms each of which is equivalent to ∥ · ∥_{D_{q}} is

(9)  ⫴f⫴_{D_{q}}^{2} := ∫_{𝔹} |D_{s}^{t}f(z)|^{2}(1 − |z|^{2})^{q+2t} dν(z),

where ν is the normalized volume measure on 𝔹. Setting dν_{q}(z) := (1−|z|^{2})^{q} dν(z), equivalently f ∈ D_{q} if and only if f ∈ H(𝔹) and D_{s}^{t}f ∈ L^{2}(ν_{q+2t}) for some s, t with q + 2t > −1, where L^{p} denotes the Lebesgue classes. For Bergman Hilbert spaces, q > −1, we take t = 0 and obtain the usual integral norms of these spaces as

⫴f⫴_{A_{q}^{2}}^{2} := ∫_{𝔹} |f(z)|^{2} dν_{q}(z) (q > −1).

The Hardy space also has an equivalent norm which is the well-known

⫴f⫴_{H^{2}}^{2} := ∫_{𝕊} |f(z)|^{2} dσ(z),

where σ is the normalized surface measure on 𝕊. Each integral norm on every D_{q} also has an accompanying integral inner product.

2.2. Möbius Transformations. Following [R, Chapter 2], the Möbius transformation that exchanges 0 and 0 ≠ a ∈ 𝔹 is the map

φ_{a}(z) := \frac{a − P_{a}(z) − \sqrt{1 − |a|^{2}}(I − P_{a})(z)}{1 − ⟨z, a⟩} (z ∈ \overline{𝔹}),

where P_{a}(z) := ⟨z, a⟩a/|a|^{2} is the projection on the complex line passing through 0 and a. It reduces to φ_{a}(z) = (a − z)/(1 − \overline{a}z) for a, z ∈ 𝔻 when N = 1. Each φ_{a} is an involution, that is, φ_{a}^{−1} = φ_{a}. An extremely useful identity for φ_{a} is

(10)  1 − ⟨φ_{a}(z), φ_{a}(w)⟩ = \frac{(1 − |a|^{2}) (1 − ⟨z, w⟩)}{(1 − ⟨z, a⟩) (1 − ⟨a, w⟩)}  (z, w ∈ \overline{𝔹}).

The complex Jacobian of φ_{a}(z) is det φ′_{a}(z) and equals

Jφ_{a}(z) = γ(z)(\frac{1 − |φ_{a}(z)|^{2}}{1 − |z|^{2}} )^{(1+N)/2} = γ(z)(\frac{1 − |a|^{2}}{|1 − ⟨z, a⟩|^{2}} )^{(1+N)/2} (z ∈ \overline{𝔹})

for some γ(z) ∈ ℂ with |γ(z)| = 1, where in obtaining the second form, (10) is used. Its real Jacobian is

J_{ℝ}φ_{a}(z) = |Jφ_{a}(z)|^{2} > 0.

We need two changes of variables formulas involving ψ ∈ M. Let G ⊂ 𝔹 and Q ⊂ 𝕊 be Borel sets, f ∈ L^{1}(ν_{q}), and F ∈ L^{1}(σ). The first is the usual

(11)  ∫_{G} f dν = ∫_{ψ^{−1}(G)} f(ψ(w))J_{ℝ}ψ(w) dν(w).

The less common second one is obtained by explicitly writing [R, p. 45, (5)] using (10) and is

(12)  ∫_{Q} F dσ = ∫_{ψ^{−1}(Q)} F (ψ(η))(J_{ℝ}ψ(η))^{\frac{N}{1+N}} dσ(η).

Let also U denote the group of all unitary transformations of ℂ^{N}. All U ∈ U are characterized by ⟨Uz, Uw⟩ = ⟨z, w⟩. If ψ ∈ M and a = ψ^{−1}(0), then there is a unique U ∈ U such that

(13)  ψ(z) = U(φ_{a}(z))  (z ∈ 𝔹).

Since JU ∈ ℂ with |JU| = 1, we see that Jψ has the same form as Jφ_{a} with a (possibly) different γ̃(z) in place of γ(z).

If U ∈ U, then

(14)  φ_{a} = U^{−1}φ_{Ua}U;

this is [CM, Lemma 2.71]. This is useful, because U acts on 𝕊 transitively and we can choose U in such way that Ua has only the first component nonzero and real. The automorphism that maps such a Ua to 0 is especially simple. For example, we use φ_{r}(z) = −φ_{−b}(z) with b = (r, 0, …, 0) and 0 < r < 1 that has the explicit form

(15)  φ_{r}(z) := (\frac{r + z_{1}}{1 + rz_{1}} , \frac{\sqrt{1 − r^{2}}}{1 + rz_{1}} z′)  (z ∈ 𝔹),

where z = (z_{1}, z′) and z′ denotes the remaining N −1 components; see [CM, p. 98]. This φ_{r} has exactly 2 fixed points, e_{1} = (1, 0, …, 0) and −e_{1}, both on 𝕊 and none in 𝔹.

Möbius transformations map balls onto ellipsoids. We need the ellipsoids described in [CM, p. 103] given by

E(e_{1}, u) := { z ∈ 𝔹 : |1 − ⟨z, e_{1}⟩|^{2} ≤ u(1 − |z|^{2}) }

with u > 0. Equivalently, z ∈ E(e_{1}, u) if and only if

|z_{1} − \frac{1}{1 + u}|^{2} + \frac{u}{1 + u} |z′|^{2} < (\frac{u}{1 + u})^{2}.

The ellipsoid E(e_{1}, u) lies in 𝔹, has center e_{1}/(1 + u), and is tangent to 𝕊 at e_{1}.

3. Proof of Main Result

We prove Theorem 1.4 and a corollary to it, and make some further comments.

Proof of Theorem 1.4. We follow the proof of [M, Theorem 1] with many detailed modifications to adapt it to several complex variables. For ψ ∈ M, note that C_{ψ^{−1}}(C_{ψ}f(z)) = C_{ψ^{−1}}f(ψ(z)) = f(ψ(ψ^{−1}(z))) = f(z) and hence C_{ψ}^{−1} = C_{ψ^{−1}}.

First we look at the case of weighted Bergman spaces. But the initial stages of the proof work for q > −(1 + N) and that is what we assume for now. Let f ∈ D_{q}. By (13) and (2), T_{ψ}^{q} = M_{θ_{ψ}^{q}}C_{φ_{a}}C_{U} for some a ∈ 𝔹 and U ∈ U. By the remarks following (13), also

βT_{ψ}^{q} = M_{θ_{φa}^{q}} C_{φ_{a}}C_{U}

for some β ∈ ℂ with |β| = 1. By a simple computation with matrices, each U ∈ U carries a monomial z^{α} to a homogeneous polynomial of the same degree |α|. Consequently C_{U} preserves parity. Thus f and f_{1} = C_{U}f have the same parity, and also g = T_{ψ}^{q}f and g_{1} = βT_{ψ}^{q}f have the same parity that is opposite to that of f_{1} by hypothesis. So without loss of generality we can replace T_{ψ}^{q} by T_{φ_{a}}^{q} and it suffices to consider

T_{φ_{a}}^{q} = M_{θ_{φa}^{q}} C_{φ_{a}}.

The fact that φ_{a}^{−1} = φ_{a} implies C_{φ_{a}}^{−1} = C_{φ_{a}} and C_{φ_{a}}^{2} = I. We have

(T_{φ_{a}}^{q})^{2}f(z) = T_{φ_{a}}^{q}((Jφ_{a}(z))^{1+ \frac{q}{1+N}} f(φ_{a}(z)))
= (Jφ_{a}(z))^{1+ \frac{q}{1+N}} (Jφ_{a}(φ_{a}(z)))^{1+ \frac{q}{1+N}} f(φ_{a}(φ_{a}(z))).

Since φ_{a}(φ_{a}(z)) = z, by the chain rule, φ′_{a}(φ_{a}(z))φ′_{a}(z) = I. Taking determinants give Jφ_{a}(φ_{a}(z))Jφ_{a}(z) = 1. This shows that (T_{φ_{a}}^{q})^{2}f(z) = f(z) and (T_{φ_{a}}^{q})^{2} = I. Setting g = T_{φ_{a}}^{q}f gives f = T_{φ_{a}}^{q}g. Thus the case f ∈ D_{q} having even parity and T_{φ_{a}}^{q}g ∈ D_{q} having odd parity coexists with the case g ∈ D_{q} having even parity and T_{φ_{a}}^{q}f ∈ D_{q} having odd parity. So it does not matter which case is investigated; let’s assume the former and keep the notation g = T_{φ_{a}}^{q}f,

Let V (z) = −z, which is unitary. Then C_{V} f = f, C_{V} g = −g, and C_{V}^{−1} = C_{V}. Then also g = −C_{V} T_{φ_{a}}^{q}f and f = C_{V} T_{φ_{a}}^{q}g. Therefore

(16)  f = −(C_{V} T_{φ_{a}}^{q})^{2}f;

that is, −1 is an eigenvalue of (C_{V} T_{φ_{a}}^{q})^{2} with f as the eigenvector. By the spectral mapping theorem, +i or −i is an eigenvalue of C_{V} T_{φ_{a}}^{q} with f as the eigenvector. But

(17)  C_{V} T_{φ_{a}}^{q}f(z) = (T_{φ_{a}}^{q}f)(−z) = (Jφ_{a}(−z))^{1+ \frac{q}{1+N}} f(φ_{a}(−z)).

Set μ_{a}(z) := φ_{a}(−z) = −φ_{−a}(z). Then μ_{a} ∈ M, μ′_{a}(z) = φ′_{a}(−z)(−I), and Jμ_{a}(z) = (−1)^{N}Jφ_{a}(−z). Hence

T_{μ_{a}}^{q}f(z) = (Jμ_{a}(z))^{1+ \frac{q}{1+N}} f(μ_{a}(z)) = (−1)^{N(1+ \frac{q}{1+N} )}(Jφ_{a}(−z))^{1+ \frac{q}{1+N}} f(φ_{a}(−z))
= (−1)^{N(1+ \frac{q}{1+N} )}C_{V} T_{φ_{a}}^{q}f(z)

using (17), and

(T_{μ_{a}}^{q})^{2}f(z) = (−1)^{2N(1+ \frac{q}{1+N} )}(C_{V} T_{φ_{a}}^{q})^{2}f(z) = (−1)^{1+2N(1+ \frac{q}{1+N} )}f(z) = κf(z)

using (16), where

κ = (−1)^{1+2N(1+ \frac{q}{1+N} )}

and |κ| = 1. Thus κ is an eigenvalue of (T_{μ_{a}}^{q})^{2}, and +\sqrt{κ} or −\sqrt{κ} is an eigenvalue of T_{μ_{a}}^{q}, both with eigenvector f. Clearly also | ± \sqrt{κ}| = 1.

By (14), μ_{a} = −φ_{−a} = −U^{−1}φ_{U(−a)}U = U^{−1}(−φ_{−U(a)})U. Choosing U ∈ U such that b := U(a) = (r, 0, …, 0) with 0 < r < 1, we obtain μ_{a} = U^{−1}φ_{r}U, where φ_{r} is as in (15). Let η = (det U)^{1+ \frac{q}{1+N}}. For f ∈ D_{q}, we have T_{U^{−1}}^{q}f(z) = \overline{η}f(U^{−1}z) and T_{U}^{q}T_{U^{−1}}^{q}f(z) = η\overline{η}f(UU^{−1}z) = f(z); hence (T_{U}^{q})^{−1} = T_{U^{−1}}^{q}. Further, we compute that T_{φ_{r}}^{q}T_{U^{−1}}^{q}f(z) = \overline{η}(Jφ_{r}(z))^{1+ \frac{q}{1+N}} f(U^{−1}φ_{r}(z)) and

T_{U}^{q}T_{φ_{r}}^{q}T_{U^{−1}}^{q}f(z) = η\overline{η}(Jφ_{r}(Uz))^{1+ \frac{q}{1+N}} f(U^{−1}φ_{r}(Uz))
= (Jφ_{r}(Uz))^{1+ \frac{q}{1+N}} f(μ_{a}(z))
= (Jμ_{a}(z))^{1+ \frac{q}{1+N}} f(μ_{a}(z)) = T_{μ_{a}}^{q}f(z),

where the equality before the last one can be seen by evaluating Jμ_{a}(z) using the chain rule. In other words, T_{φ_{r}}^{q} = (T_{U}^{q})^{−1}T_{μ_{a}}^{q}T_{U}^{q}. Since a similarity transformation preserves eigenvalues and eigenvectors, we conclude that +\sqrt{κ} or −\sqrt{κ} is an eigenvalue of T_{φ_{r}}^{q} with eigenvector f ∈ D_{q}.

We have lim_{z→e_{1}} φ_{r}(z) = e_{1} and let

δ := lim_{z→e_{1}} \frac{1 − |φ_{r}(z)|}{1 − |z|} = lim_{z→e_{1}} \frac{1 − |φ_{r}(z)|^{2}}{1 − |z|^{2}} ,

where the limits are unrestricted from within 𝔹. A quick computation shows that δ = 1/(1 + r) < 1. By [CM, Lemma 2.77] due to Julia, φ_{r}(E(e_{1}, u)) ⊂ E(e_{1}, δu), and by [CM, Proposition 2.85], E(e_{1}, δu) ⊂ E(e_{1}, u). Together we have the inclusion φ_{r}(E(e_{1}, u)) ⊂ E(e_{1}, u).

For n = 1, 2, …, denote the forward iterates of φ_{r} by φ_{r}^{n} = φ_{r} ∘ φ_{r}^{n−1}, where φ_{r}^{0} is the identity, and its backward iterates by φ_{r}^{−n} = (φ_{r}^{n})^{−1}. By the properties on the ellipsoids E(e_{1}, u) and of φ_{r} and the Denjoy-Wolff theorem, as n → ∞, φ_{r}^{n} converges uniformly on compact subsets of 𝔹 to e_{1}; see [CM, Theorem 2.83 and Proposition 2.88]. In other words, e_{1} is the attracting fixed point of φ_{r} and its Denjoy-Wolff point. Now fix u = 1, call the corresponding E(e_{1}, 1) =: E, and let G = E \φ_{r}(E), which is nonempty by above. As a consequence of all the discussion about the ellipsoids, for any 0 < r < 1 we have

(18)  𝔹 = ⋃_{n∈ℤ} φ_{r}^{n}(G)

and the sets φ_{r}^{n}(G) for different n’s are disjoint.

In the remaining part of the proof we first restrict to q > −1 for which D_{q} = A_{q}^{2}, weighted Bergman spaces. Now applying the change of variables z = φ_{r}(w), using (11) and that f is an eigenvector yield

∫_{φ_{r}^{n+1}(G)} |f(z)|^{2} dν_{q}(z) = ∫_{φ_{r}^{n}(G)} |f(φ_{r}(w))|^{2}(1 − |φ_{r}(w)|^{2})^{q}J_{ℝ}φ_{r}(w) dν(w)

= ∫_{φ_{r}^{n}(G)} |f(φ_{r}(w))|^{2} \frac{(1 − |φ_{r}(w)|^{2})^{1+N+q}}{(1 − |w|^{2})^{1+N}} dν(w)

= ∫_{φ_{r}^{n}(G)} |f(φ_{r}(w))|^{2}|(Jφ_{r}(w))^{1+ \frac{q}{1+N}} |^{2}(1 − |w|^{2})^{q} dν(w)

= ∫_{φ_{r}^{n}(G)} |T_{φ_{r}}^{q}f(w)|^{2} dν_{q}(w) = ∫_{φ_{r}^{n}(G)} |± \sqrt{κ}f(w)|^{2} dν_{q}(w)

= ∫_{φ_{r}^{n}(G)} |f(z)|^{2} dν_{q}(z).

Thus the above integrals have the same value on all the sets φ_{r}^{n}(G) for n ∈ ℤ which is equal to the value of the integral on φ_{r}^{0}(G) = G. But f is an eigenvector and hence is not the zero function, and since f ∈ H(𝔹), none of the integrals on the φ_{r}^{n}(G) is 0. On the other hand, f ∈ D_{q} and hence ⫴f⫴_{D_{q}} < ∞. But by (18) we also have

⫴f⫴_{D_{q}}^{2} = ∫_{𝔹} |f(z)|^{2} dν_{q}(z) = ∑_{n∈ℤ} ∫_{φ_{r}^{n}(G)} |f(z)|^{2} dν_{q}(z) = ∑_{n∈ℤ} ∫_{G} |f(z)|^{2} dν_{q}(z) = ∞.

This contradiction shows that a nonzero f having the parity properties in the statement of the theorem cannot exist for q > −1.

Next we take care of the case q = −1, the Hardy space. Let D be the intersection of the ellipsoid E = E(e_{1}, 1) with the complex line [e_{1}] through 0 and e_{1}, which is given by |z_{1} − 1/2| < 1/4. The set G̃ = D \ φ_{r}(D) is nonempty just like G ≠ ∅. Let also Q = { (z_{1}, z′) : z_{1} ∈ G̃, |z_{1}|^{2} +|z′|^{2} = 1 }; this is that part of 𝕊 that lies “above” G̃. We have 𝔹∩[e_{1}] = ⋃_{n∈ℤ} φ_{r}^{n}(G̃) and 𝕊 = { (z_{1}, z′) : z_{1} ∈ 𝔻, |z_{1}|^{2}+|z′|^{2} = 1 }, the second modulo a set of σ-measure 0. Then just like (18), we have 𝕊 = ⋃_{n∈ℤ} φ_{r}^{n}(Q) modulo a set of σ-measure 0, which is a disjoint union.

Now we apply the change of variables ζ = φ_{r}(η), use (12) and that f is an eigenvector to obtain

∫_{φ_{r}^{n+1}(Q)} |f(ζ)|^{2} dσ(ζ) = ∫_{φ_{r}^{n}(Q)} |f(φ_{r}(η))|^{2}|(Jφ_{r}(η))^{1− \frac{1}{1+N}} |^{2} dσ(η)
= ∫_{φ_{r}^{n}(Q)} |T_{φ_{r}}^{−1}f(η)|^{2} dσ(η) = ∫_{φ_{r}^{n}(Q)} |± \sqrt{κ}f(η)|^{2} dσ(η)
= ∫_{φ_{r}^{n}(Q)} |f(ζ)|^{2} dσ(ζ).

As in the case q > −1, each integral on φ_{r}^{n}(Q) can be replaced by one on Q. But f ∈ H^{2} is an eigenvector, so is not the zero function, and by [R, Theorem 5.6.4 (b)], its boundary values on 𝕊 are nonzero σ-a.e.. Then none of the integrals on the φ_{r}^{n}(Q) is 0. On the other hand, f ∈ H^{2} and hence ⫴f⫴_{H^{2}} < ∞. Similar to the case q > −1, we have

⫴f⫴_{H^{2}}^{2} = ∫_{𝕊} |f(ζ)|^{2} dσ(ζ) = ∑_{n∈ℤ} ∫_{φ_{r}^{n}(Q)} |f(ζ)|^{2} dσ(ζ) = ∑_{n∈ℤ} ∫_{Q} |f(ζ)|^{2} dσ(ζ) = ∞.

By this contradiction, the theorem is proved for the case q = −1 too.

Lastly, we consider the case q < −1. Pick s, t ∈ ℝ such that p = q + 2t > −1. Here we prove the result not for T_{ψ}^{q} but for Y_{ψ}^{q} = D_{s+t}^{−t}T_{ψ}^{p}D_{s}^{t} where T_{ψ}^{p} : A_{p}^{2} → A_{p}^{2}. By (8) and (7), we have Y_{ψ}^{q} : D_{q} → D_{q}. We have also T_{ψ}^{p} = D_{s}^{t}Y_{ψ}^{q}D_{s+t}^{−t} and that Y_{ψ}^{q} and T_{ψ}^{p} are similar operators. Now suppose, without loss of generality, that f ∈ D_{q} has even parity and Y_{ψ}^{q}f has odd parity. By their very definitions, the D_{s}^{t} preserve parity and g = D_{s}^{t}f ∈ A_{p}^{2} also has even parity. On the other hand, since f = D_{s+t}^{−t}g, we see that T_{ψ}^{p}g = D_{s}^{t}Y_{ψ}^{q}f has odd parity. By the already proved case for the Bergman space A_{p}^{2}, we conclude that g = 0. Then also f = 0. In fact, Y_{ψ}^{q}f = λf if and only if D_{s+t}^{−t}T_{ψ}^{p}g = λf if and only if T_{ψ}^{p}g = λg for some λ ∈ ℂ.

The proof of Theorem 1.4 is now complete. □

Remark 3.1. The change of variables that is used in transformung the integrals in the case q > −1 can be expressed in the form that the measures ν_{q} are invariant

under the transformations Z_{ψ}^{q}f(z) := f(ψ(z))(Jψ(z))^{2(1+ \frac{q}{1+N} )} in the sense that

∫_{𝔹} Z_{ψ}^{q}f dν_{q} = ∫_{𝔹} f dν_{q} (f ∈ L^{1}(ν_{q}), q ∈ ℝ).

This is already noted in [K, (21)].

Remark 3.2. The last part of the proof involving integrals does not work for q < −1, because a positive-order derivative on f is required in the integral norms of D_{q} on 𝔹 for all q < −1; see [KÜ, Corollary 7.2]. So, for example, if we pick t so that q + 2t = 0 for simplicity in (9), we end up with an integral of |T_{φ_{r}}^{0}D_{s}^{t}f|^{2} on φ_{r}^{n}(G). For the proof to go through, we need D_{s}^{t}f ∈ A_{0}^{2} to be an eigenvector of T_{φ_{r}}^{0}. This would be implied by T_{φ_{r}}^{0}D_{s}^{t}f having odd parity when f and hence D_{s}^{t}f have even parity. But what we know is that T_{φ_{r}}^{q}f has odd parity and this need not imply that T_{φ_{r}}^{0}D_{s}^{t}f has odd parity because of the differences between T_{ψ}^{q} and T_{ψ}^{0}.

Such differences do not prevent [M] from obtaining the theorem for the Dirichlet space, because when N = 1, the first-order ordinary derivative and the chain rule are enough to move between that space and the unweighted Bergman space. Neither of these tools is available for N ≥ 2. These are exactly the reasons why we resort to the other operators Y_{ψ}^{q} when q < −1.

Remark 3.3. The value κ depends in general on both N and q. For the unweighted Bergman space, q = 0, κ = (−1)^{1+2N} = −1, and the eigenvalues that are shown not to exist in the proof of Theorem 1.4 are +\sqrt{κ} = +i and −\sqrt{κ} = −i independently

of dimension N. For the Hardy space, q = −1 and κ = (−1)^{\frac{1+N+2N^{2}}{1+N}}. If also N = 1, κ = (−1)^{2} = +1 and the eigenvalues that are shown not to exist in the proof of Theorem 1.4 are +\sqrt{κ} = +1 and −\sqrt{κ} = −1, contrary to what is claimed in the proof of [M, Theorem 1]. But as already noted in [M, Remark 1], the proof of [M, Theorem 1] as well as of Theorem 1.4 here depend only on | ± \sqrt{κ}| = 1 and are unaffected.

Corollary 3.4. Let q ≥ −1 and ψ ∈ M. There is a nonzero function f ∈ D_{q} such that f and T_{ψ}^{q}f have the same parity if and only if ψ = U ∈ U. For q < −1, the same result holds for the operators Y_{ψ}^{q}.

Proof. Let q ≥ −1 first. If ψ = U ∈ U, since compositions with U and multiplication with complex numbers that are Jacobians of such composition operators preserve parity, there are f as claimed.

Conversely, let ψ = φ_{a} in which a ≠ 0 and suppose an f as claimed exists. If we repeat the proof of Theorem 1.4 carefully considering the case f even and T_{φ_{a}}^{q}f even and the case f odd and T_{φ_{a}}^{q}f odd, we obtain f = (C_{V} T_{φ_{a}}^{q})^{2}f instead of (16). This in turn yields two complex numbers of modulus 1 one of which is an eigenvalue of T_{μ_{a}}^{q} and of T_{φ_{r}}^{q} with f as an eigenvector. In the rest of the proof, the only property of the eigenvalues used is that they are of modulus 1. Again we conclude that f = 0.

The case q ≤ −1 is automatic since it depends on the conclusion of the case q > −1. □

Acknowledgments

The author thanks A. Ersin Üreyen of Eskişehir Technical University for useful discussions and an anonymous referee for careful attention to details.

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Bi̇lkent Üni̇versi̇tesi̇, Matemati̇k Bölümü, 06800 Ankara

kaptan@fen.bilkent.edu.tr
http://www.fen.bilkent.edu.tr/∼kaptan/