UNCERTAINTY PRINCIPLES IN HOLOMORPHIC FUNCTION SPACES ON THE UNIT BALL

H. TURGAY KAPTANOĞLU

Abstract. On all Bergman-Besov Hilbert spaces on the unit disc, we find self-adjoint weighted shift operators that are differential operators of half order whose commutators are the identity thereby obtaining uncertainty relations in these spaces. We also obtain joint average uncertainty relations for pairs of commuting tuples of operators on the same spaces defined on the unit ball. We further identify functions that yield equality in some uncertainty inequalities.

1. Introduction

The uncertainty principle of Heisenberg originates in quantum physics. The fact that quantum theory is based on operators on Hilbert spaces avails oneself of the consideration of uncertainty principles as inequalities involving Hilbert space operators.

Theorem 1.1. Let L and M be self-adjoint operators on a Hilbert space H with inner product ⟨· , ·⟩_{H} and the associated norm ∥ · ∥_{H}. Then

(1)  ∥(L − λI)u∥_{H} ∥(M − μI)u∥_{H} ≥ \frac{1}{2}|⟨(LM − ML)u, u⟩_{H} |

for all λ, μ ∈ ℝ and all u that lie in the domain of both LM and ML. Equality holds if and only if (L − λI)u = iγ(M − μI)u for some γ ∈ ℝ.

Mathematically, as soon as self-adjoint operators L and M are found on a Hilbert space H whose commutator [L, M] := LM −ML is a scalar multiple of the identity operator whence the right side of (1) simplifies (the case of conjugate observables), an explicit uncertainty principle appears. Theorem 1.1 and its easy proof can be found in [F] and [G], coincidentallly on pp. 27–28 in both.

A few remarks are in order. The equality [L, M] = cI for some constant c cannot be satisfied with both L and M bounded.

Theorem 1.2. If Z is a Banach algebra with unit element e and if x, y ∈ Z, then xy − yx ≠ e.

Theorem 1.2 can be found in [R, Theorem 13.6] and its proof does not even use the completeness of Z. So, as warned in [FS, p. 211], the intersection of the domains of LM and ML is crucial even when [L, M] = cI on the intersection.

Date: 4 July 2023.

2020 Mathematics Subject Classification. Primary 81S07; Secondary 26A33, 30B10, 30H20, 30H25, 32A05, 32A36, 46E20, 46E22, 46N50, 47B32, 47B37.

Key words and phrases. Uncertainty principle, weighted shift operator, annihilation operator, creation operator, number operator, radial derivative, bosonic Fock space, Bergman-Besov space, Hardy space, Dirichlet space, Drury-Arveson space.

Using the mathematical approach above, uncertainty principles with [L, M] = cI are obtained in the Segal-Bargmann-Fischer-Fock space of entire functions weighted with the Gaussian in [CZ] or in its generalizations in [L]. Further, [CZ] poses the problem of finding uncertainty principles in Hardy and Bergman spaces. Some results are presented in [So2, So1] on certain, but not all, Bergman and Dirichlet spaces on the unit disc, but in these sources, one of the operators is always taken as the first-order derivative resulting in [L, M] ≠ cI. It is shown in [CD, Theorem 11] that there are no first order self-adjoint differential operators on weighted Bergman spaces on the unit disc whose commutator is a nonzero multiple of the identity.

We find self-adjoint operators L and M with LM − ML = cI on a large family of weighted symmetric (bosonic) Fock spaces of holomorphic functions on the unit disc as studied in [Ka] and obtain uncertainty relations from them. This family includes all Hilbert spaces among Bergman-Besov spaces, Dirichlet spaces, and the Hardy space H^{2}. The operators L and M are combinations of specific weighted shift operators, and these shifts are fractional differential operators of order 1/2 and also nothing but annihilation and creation operators. For contrast, uncertainty relations are obtained in [UT] in which the operators are annihilation and number operators. We also obtain joint average uncertainty relations for pairs of commuting tuples of operators in the same family of Fock spaces on the unit ball in ℂ^{n} which includes the Drury-Arveson space. The formulation with tuples of operators seems new.

Our main results are Theorem 4.3 and Theorem 5.2.

2. Notation and Preliminaries

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}, where we conjugate the second variable following the tradition in mathematics, and the associated norm |z| = \sqrt{⟨z, z⟩}. When n = 1, the ball is the unit disc 𝔻 in the complex plane.

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}}. We let ϵ_{j} := (0, …, 0, 1, 0, …, 0), where 1 is in the jth position and the right side of := defines its left side.

The Pochhammer symbol (p)_{q} is defined by

(p)_{q} = \frac{Γ(p + q)}{Γ(p)}

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

(2) \frac{Γ(r + p)}{Γ(r + q)} ∼ r^{p−q},  \frac{(p)_{r}}{(q)_{r}} ∼ r^{p−q},  \frac{(r)_{p}}{(r)_{q}} ∼ r^{p−q}  (Re r → ∞),

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

Definition 2.1. A function K(z, w) is called the reproducing kernel of a Hilbert space H of functions defined on 𝔹 if K(z, ·) ∈ H for each z ∈ 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. We deal with Hilbert function spaces whose elements are holomorphic functions on the unit ball, the collection of all of which we denote by H(𝔹).

We use the term operator to mean a linear transformation whose domain D(T) and range R(T) are subspaces of a complex Hilbert space H with no requirement on boundedness.

If C is a densely defined operator on H, denoting its adjoint A := C^{∗}, then by [R, Theorem 13.9], A is a closed operator. If A is also densely defined, then by [R, Theorem 13.12], A^{∗∗} = A. Further, by [Kr, Theorem 10.2-1], C ⊂ C^{∗∗} = A^{∗}, that is, C = A^{∗} on the domain of C. If we let L := C + C^{∗} and M := i(C − C^{∗}), then L and M are self-adjoint and

(3)  [L, M] = 2i[A, C].

Moreover, if λ, μ ∈ ℝ, then L − λI and M − μI are also self-adjoint and

(4)  [L − λI, M − μI] = [L, M].

Self-adjointness, (3), and (4) hold on the intersection of the domains of AC and CA, which is included in the intersection of the domains of C and A.

Two abstract uncertainty principles that follow from Theorem 1.1 and given in [F, pp. 27–28] are the following.

Corollary 2.2. Let C and A be densely defined operators on a Hilbert space H with the properties that A = C^{∗} and [A, C] = I.

(i) ∥(C + A − λI)u∥_{H}∥(C − A − iμI)u∥_{H} ≥ ∥u∥_{H}^{2} for all λ, μ ∈ ℝ and all u that lie in the domain of both CA and AC.

(ii) ∥(C + A)u∥_{H}^{2} + ∥(C − A)u∥_{H}^{2} ≥ 2∥u∥_{H}^{2} for all u that lie in the domain of both CA and AC.

The passage from (i) to (ii) is via the elementary inequality a^{2} + b^{2} ≥ 2ab, so we do not dwell on (ii) any more. Inequality (1) can be generalized to hold also for complex λ and μ as explained in [CZ]. However, for equality, λ and μ must be real. This generalization works for any pairs of operators, and we do not dwell on this any more either.

If T = (T_{1}, …, T_{n}) and S = (S_{1}, …, S_{n}) are tuples of operators on the same Hilbert space H, we use the notation T · S := T_{1}S_{1} + ⋯ + T_{n}S_{n}. We define the commutator of the tuples T and S by [T, S] := T · S − S · T. With these definitions, if τI = (τ_{1}I, …, τ_{n}I) with τ_{j} ∈ ℂ, j = 1, …, n, and σI is similar, then by a straightforward calculation,

(5)  [T − τI, S − σI] = [T, S].

3. Weighted Symmetric Fock Spaces

In [Ka], large families of weighted symmetric (bosonic) Fock spaces of holomorphic functions on 𝔹 are studied following [A]. They are the spaces in which we develop uncertainty principles. The material in this section is taken from [Ka].

Definition 3.1. Let b := (b_{k})_{k} be a weight sequence satisfying b_{0} = 1, b_{k} > 0 for all k = 0, 1, 2, …, and

(6)  lim sup_{k→∞} b_{k}^{1/k} ≤ 1.

4 H. TURGAY KAPTANOĞLU

We define positive definite kernels by

(7)  K_{b}(z, w) := ∑_{k=0}^{∞} b_{k}⟨z, w⟩^{k} = ∑_{k=0}^{∞} b_{k} ∑_{|α|=k} \frac{|α|!}{α!}z^{α}\overline{w}^{α}  (z, w ∈ 𝔹)

and spaces F_{b} as the reproducing kernel Hilbert spaces generated by these kernels.

Condition (6) causes the series in (7) to converge absolutely and uniformly for (z, w) in compact subsets of 𝔹 × 𝔹 thereby defining K_{b} as a holomorphic function of z ∈ 𝔹 and a conjugate holomorphic function of w ∈ 𝔹. Theorem 3.2. The space F_{b} consists of all f ∈ H(𝔹) with Taylor expansions

(8)  f(z) = ∑_{|α|=0}^{∞} f_{α}z^{α}

converging absolutely and uniformly on compact subsets of 𝔹 for which

(9)  ∥f∥_{b}^{2} := ∑_{|α|=0}^{∞} \frac{1}{b_{|α|}} \frac{α!}{|α|!}|f_{α}|^{2} < ∞

and is equipped with the inner product

⟨f, g⟩_{b} := ∑_{|α|=0}^{∞} \frac{1}{b_{|α|}} \frac{α!}{|α|!}f_{α}\overline{g}_{α}.

Further,

(10)  B_{b} := \{ e_{α}^{b}(z) := \sqrt{b_{|α|} \frac{|α|!}{α!}} z^{α} : α ∈ ℕ^{n} \}

is an orthonormal basis for F_{b}. Moreover, holomorphic polynomials in the n variables z_{1}, …, z_{n} are dense in each F_{b}.

In particular, for each α ∈ ℕ^{n},

∥z^{α}∥_{b}^{2} = \frac{1}{b_{|α|}} \frac{α!}{|α|!}.

We now describe a particular family of holomorphic kernels and associated Hilbert function spaces that include many well-known spaces as special cases. They are included in the family of Bergman-Besov spaces on 𝔹. Definition 3.3. For q ∈ ℝ and k = 0, 1, 2, …, we set

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}

denote by K_{q}(z, w) := ∑_{k=0}^{∞} b_{k}(q)⟨z, w⟩^{k} the reproducing kernel with coefficient sequence (b_{k}(q))_{k} and by F_{q} the Hilbert space generated by the kernel K_{q}.

By (2),

(11)  b_{k}(q) ∼ k^{n+q}  (k → ∞)

for any q ∈ ℝ assuring that (6) is satisfied. So an f ∈ H(𝔹) given by (8) belongs to F_{q} if and only if

(12)  ∑_{|α|=1}^{∞} \frac{1}{|α|^{n+q}} \frac{α!}{|α|!} |f_{α}|^{2} < ∞.

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}

where _{2}F_{1} is the Gauss hypergeometric function. In particular,

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

Thus F_{q} is the 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). We simply write A^{2} for the unweighted Bergman space when q = 0. If q < −(1 + n), then the functions in F_{q} are bounded on 𝔹. The inner products and hence the norms of all the spaces in the F_{q} family can be expressed as integrals on 𝔹 of either the functions or their sufficiently high-order derivatives; see [Ka] for details.

4. Uncertainty Principles in Spaces on the Disc

We start with the case of the function spaces on the unit disc, that is, n = 1. Many of the formulas in Section 3 are simplified mainly because now |α| = α = k. So the terms of the orthonormal basis B_{b} of F_{b} are e_{k}^{b}(z) = \sqrt{b_{k}} z^{k} for k = 0, 1, 2, …; in particular, e_{0}^{b} = 1. Equivalently,

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

The homogeneous expansion of a function f ∈ H(𝔻) is now its Taylor expansion which can also be written in terms of the orthonormal basis of F_{b} as

(14)  f(z) = ∑_{k=0}^{∞} f_{k}z^{k} = ∑_{k=0}^{∞} \frac{f_{k}}{\sqrt{b_{k}}} e_{k}^{b}(z)  (z ∈ 𝔻).

Definition 4.1. We define the operator C_{b} on F_{b} by first letting

C_{b}e_{k}^{b} := \sqrt{k + 1} e_{k+1}^{b}

on B_{b} and then by extending it linearly to its span.

Thus the operator C_{b} on F_{b} is nothing but the weighted shift operator with the weight sequence (\sqrt{k + 1})_{k}. Such operators are investigated in detail in [Sh]. Since the weight sequence is unbounded, C_{b} is unbounded on F_{b}. However, it is densely defined since polynomials are dense in every F_{b}.

By [Sh] or in a similar way to the unweighted shifts on F_{b} studied in [Ka], the adjoint A_{b} := C_{b}^{∗} of C_{b} is given by

A_{b}e_{k}^{b} = \sqrt{k} e_{k−1}^{b} (k ≥ 1)

and A_{b}e_{0}^{b} = 0. Similarly, A_{b} is densely defined and unbounded on F_{b}, and A_{b}^{∗} = C_{b}. The choice of the letters C and A is no coincidence, because these operators are exactly the creation and annihilation operators on many-body quantum systems; see [T, p. 106] or [MR, p. 92]. If f ∈ F_{b} is given by (14), then

\begin{aligned}C_{b}f(z) & = ∑_{k=0}^{∞} \sqrt{k + 1} \sqrt{\frac{b_{k+1}}{b_{k}}} f_{k}z^{k+1}, \\ A_{b}f(z) & = ∑_{k=1}^{∞} \sqrt{k} \sqrt{\frac{b_{k−1}}{b_{k}}} f_{k}z^{k−1}. \end{aligned} ; (15)

Continuing,

A_{b}C_{b}e_{k}^{b} = (k + 1)e_{k}^{b}  and  C_{b}A_{b}e_{k}^{b} = ke_{k}^{b} (k ≥ 1).

If f ∈ F_{b}, then

\begin{aligned}C_{b}A_{b}f(z) & = ∑_{k=1}^{∞} kf_{k}z^{k} =: Nf(z), \\ A_{b}C_{b}f(z) & = ∑_{k=0}^{∞}(k + 1)f_{k}z^{k} = Nf(z) + If(z), \end{aligned} ; (16)

where N denotes the number operator of physics which is the same as the radial derivative of mathematics. Thus

(17)  (A_{b}C_{b} − C_{b}A_{b})f = f,  that is,  [A_{b}, C_{b}] = I

on the subspace of F_{b} that is the intersection of the domains if A_{b}C_{b} and C_{b}A_{b}. In fact, (\sqrt{k + 1})_{k} is the only positive weight sequence with initial term 1 such that this exact commutation relation holds. This commutation relation is well known, but in general not in reference to an uncertainty relation; see [T, p. 104]. Here we use it to formulate uncertainty relations on spaces not generally considered in quantum physics and also identify clearly the domains of the operators.

Let’s describe the domains of the operators. The domain D(C_{b}) of C_{b} is the subspace of all f ∈ F_{b} for which also C_{b}f ∈ F_{b}. So, by Theorem 3.2, (13), (15), and (16),

\begin{aligned}E_{b} & := \{ f ∈ H(𝔻) : ∑_{k=1}^{∞} \frac{k}{b_{k}} |f_{k}|^{2} < ∞ \} = D(C_{b}) = D(A_{b}), \\ G_{b} & := \{ f ∈ H(𝔻) : ∑_{k=1}^{∞} \frac{k^{2}}{b_{k}} |f_{k}|^{2} < ∞ \} = D(A_{b}C_{b}) = D(C_{b}A_{b}), \end{aligned} ; (18)

It is clear that G_{b} ⊂ E_{b} ⊂ F_{b} and each inclusion is dense since all three sets contain polynomials. By [R, Theorem 13.9], we conclude the following.

Proposition 4.2. C_{b}, A_{b}, A_{b}C_{b}, and C_{b}A_{b} are closed operators.

Let L_{b} = C_{b} +A_{b} and M_{b} = i(C_{b} −A_{b}) be the self-adjoint operators as in Section 2. Then

L_{b}f(z) = \sqrt{b_{1}} f_{0}z + ∑_{k=1}^{∞}(\sqrt{k + 1} \sqrt{\frac{b_{k+1}}{b_{k}}} z^{2} + \sqrt{k} \sqrt{\frac{b_{k−1}}{b_{k}}} )f_{k}z^{k−1}

= \frac{1}{\sqrt{b_{1}}} f_{1} + ∑_{k=1}^{∞}(\sqrt{k} \sqrt{\frac{b_{k}}{b_{k−1}}} f_{k−1} + \sqrt{k + 1} \sqrt{\frac{b_{k}}{b_{k+1}}} f_{k+1}) z^{k},

where we change variables from k + 1 to k in the first sum and from k − 1 to k in the second sum to write the second expression. We do not show M_{b}f(z) since it is so similar to L_{b}f(z). Both L_{b} and M_{b} are closed operators with domain E_{b}. By (3) and (17),

(19)  [L_{b}, M_{b}] = 2iI

on G_{b} = D([L_{b}, M_{b}]). The uncertainty principle in F_{b} implied by Corollary 2.2 (i) is the following.

Theorem 4.3. For f ∈ G_{b} ⊂ F_{b} and λ, μ ∈ ℝ,

∥(C_{b} + A_{b} − λI)f∥_{b} ∥(C_{b} − A_{b} − iμI)f∥_{b} ≥ ∥f∥_{b}^{2}.

For λ = μ = 0, equality holds for a function in G_{b} if and only if it is a complex scalar multiple of

f^{b}(z) = ∑_{l=0}^{∞}(\frac{γ + 1}{γ − 1})^{l}\sqrt{\frac{1 · 3 · 5 ⋯ (2l − 1)}{2 · 4 · 6 ⋯ 2l}} \sqrt{b_{2l}} z^{2l}

for some γ < 0. For (λ, μ) ≠ (0, 0), Taylor series coefficients of the functions that give equality can be obtained from a three-term recurrence relation.

Proof. By (19) and (4), for λ, μ ∈ ℝ,

2i⟨f, f⟩_{b} = ⟨[L_{b}, M_{b}]f, f⟩_{b} = ⟨[L_{b} − λI, M_{b} − μI]f, f⟩_{b}
= ⟨(L_{b} − λI)(M_{b} − μI)f, f⟩_{b} − ⟨(M_{b} − μI)(L_{b} − λI)f, f⟩_{b}
= ⟨(M_{b} − μI)f, (L_{b} − λI)f⟩_{b} − \overline{⟨(M_{b} − μI)f, (L_{b} − λI)f⟩_{b}}
= 2i Im⟨(M_{b} − μI)f, (L_{b} − λI)f⟩_{b}.

Hence

∥f∥_{b} ≤ |⟨(M_{b} − μI)f, (L_{b} − λI)f⟩_{b}| ≤ ∥(M_{b} − μI)f∥_{b}∥(L_{b} − λI)f∥_{b}

on G_{b}, which is the desired uncertainty inequality.

Equality holds in the first inequality if and only if ⟨M_{b}f, L_{b}f⟩_{b} is pure imaginary with positive imaginary part. Equality holds in the second inequality if and only if L_{b}f = βM_{b}f for some β ∈ ℂ. Applying the two conditions together, we deduce

that equality holds in the uncertainty inequality for an f^{b}(z) = ∑_{k=0}^{∞} f_{k}^{b}z^{k} ∈ G_{b} if

and only if L_{b}f^{b} − λf^{b} = iγ(M_{b}f^{b} − μf^{b}) for some γ < 0. Equivalently,

\begin{matrix}\frac{f_{1}^{b}}{\sqrt{b_{1}}} & − λf_{0}^{b} + ∑_{k=1}^{∞}(\sqrt{k} \sqrt{\frac{b_{k}}{b_{k−1}}} f_{k−1}^{b} + \sqrt{k+1} \sqrt{\frac{b_{k}}{b_{k+1}}} f_{k+1}^{b} − λf_{k}^{b}) z^{k} \\ = & \frac{γf_{1}^{b}}{\sqrt{b_{1}}} − iγμf_{0}^{b} + γ ∑_{k=1}^{∞}(\sqrt{k+1} \sqrt{\frac{b_{k}}{b_{k+1}}} f_{k+1}^{b} − \sqrt{k} \sqrt{\frac{b_{k}}{b_{k−1}}} f_{k−1}^{b} − iμf_{k}^{b}) z^{k}. \end{matrix} ; (20)

Setting the coefficients of z^{k} on both sides of (20) equal to each other, letting f_{−1}^{b} = 0 for convenience, and using γ < 0, we obtain

(21)  f_{k+1}^{b} = \frac{γ + 1}{γ − 1} \sqrt{\frac{k}{k + 1}} \sqrt{\frac{b_{k+1}}{b_{k−1}}} f_{k−1}^{b} − \frac{λ − iγμ}{γ − 1} \frac{1}{\sqrt{k + 1}} \sqrt{\frac{b_{k+1}}{b_{k}}} f_{k}^{b}

for k = 0, 1, 2, . . .. From this three-term recurrence relation, it is possible to determine the Taylor series coefficients of f^{b} in terms of f_{0}^{b} and f_{1}^{b} and check for what γ < 0 this f^{b} belongs to G_{b}. But the computations are cumbersome and we only work out the details of the representative case λ = μ = 0. As a side note, if γ = 1, then the only function satisfying (20) is the zero function.

For λ = μ = 0, the constant terms of (20) give

(22)  (1 − γ)f_{1}^{b} = 0.

and the recurrence relation (21) takes the form

(23)  f_{k+1}^{b} = \frac{γ + 1}{γ − 1} \sqrt{\frac{k}{k + 1}} \sqrt{\frac{b_{k+1}}{b_{k−1}}} f_{k−1}^{b}.

Since γ < 0, (22) gives f_{1}^{b} = 0. Then from (23), f_{3}^{b} = f_{5}^{b} = ⋯ = 0 as well. Letting k = 2l − 1, the coefficients with even indices satisfy

f_{2l}^{b} = (\frac{γ + 1}{γ − 1})^{l}\sqrt{\frac{1 · 3 · 5 ⋯ (2l − 1)}{2 · 4 · 6 ⋯ 2l}} \sqrt{b_{2l}} f_{0}^{b} (l = 1, 2, …).

Denoting the first square root by d_{2l} and setting d_{0} = 1, we have

(24)  f^{b}(z) = f_{0}^{b} ∑_{l=0}^{∞}(\frac{γ + 1}{γ − 1})^{l}d_{2l} \sqrt{b_{2l}} z^{2l} = f_{0}^{b} ∑_{l=0}^{∞} c_{2l}^{b}z^{2l}.

Simply d_{2l} ≤ 1/\sqrt{2} and by (18), f^{b} ∈ G_{b} if and only if

(25)  ∑_{l=1}^{∞} \frac{l^{2}}{b_{2l}} |c_{2l}^{b}|^{2} = ∑_{l=1}^{∞} l^{2}|\frac{γ + 1}{γ − 1}|^{2l}d_{2l}^{2} < ∞.

Since (d_{2l})_{l} is bounded, the finiteness in (25) is implied by the finiteness of

∑_{l=1}^{∞} l^{2}|\frac{γ + 1}{γ − 1}|^{2l},

and this holds when |γ + 1| < |γ − 1|, that is, γ < 0.

Thus the precise range of values of γ for having a nontrivial f^{b} ∈ G_{b} given in (24) for which equality holds in the uncertainty inequality for λ = μ = 0 is γ < 0. □

We next specialize to the F_{q} spaces of Definition 3.3 in which the b_{k}(q) have specific values for q ∈ ℝ, and use the subscript q as in C_{q} to denote an operator on F_{q}. We have

C_{q}z^{k} = c_{k+1}^{q}z^{k+1} = \begin{cases}\sqrt{2 + q + k} z^{k+1}, & if q > −2, \\ \frac{k + 1}{\sqrt{−q + k}} z^{k+1}, & if q ≤ −2, \end{cases}

and

A_{q}z^{k} = a_{k−1}^{q}z^{k−1} = \begin{cases}\frac{k}{\sqrt{1 + q + k}} z^{k−1} = \frac{1}{\sqrt{1 + q + k}} (z^{k})′, & if q > −2, \\ \sqrt{−q + k − 1} z^{k−1} = \frac{\sqrt{−q + k − 1}}{k} (z^{k})′, & if q ≤ −2, \end{cases}

where primes denote differentiation. These explicit formulas show that both C_{q} and A_{q} are fractional differential operators of order 1/2 since c_{k}^{q} ∼ a_{k}^{q} ∼ k^{1/2} for each q ∈ ℝ. For comparison, by (16), C_{b}A_{b} and A_{b}C_{b} are differential operators of order 1.

Three values of q give three important spaces; F_{0} is the Bergman space A^{2}, F_{−1} is the Hardy space H^{2}, F_{−2} is the Dirichlet space D. There is no distinction between the Hardy space and the Drury-Arveson space when n = 1. The precise forms of C_{q} and A_{q} on these spaces can be read off from the above formulas.

Further, for f ∈ F_{q},

L_{q}f(z) = \begin{cases}\frac{1}{\sqrt{2 + q}} f_{1} + ∑_{k=1}^{∞}(\sqrt{1 + q + k} f_{k−1} + \frac{k + 1}{\sqrt{2 + q + k}} f_{k+1})z^{k}, & if q > −2, \\ \sqrt{−q} f_{1} + ∑_{k=1}^{∞}(\frac{k}{\sqrt{−1 − q + k}} f_{k−1} + \sqrt{−q + k} f_{k+1})z^{k}, & if q ≤ −2, \end{cases}

and M_{q}f(z) is similar. So L_{q} and M_{q} are also fractional differential operators of order 1/2 for each q ∈ ℝ.

The domains of C_{q}, A_{q}, L_{q}, and M_{q} are all the same, E_{q}. A comparison of (18), (12), and (11) reveals that E_{q} = F_{q−1}. Similarly, the domain of A_{q}C_{q}, C_{q}A_{q}, and [L_{q}, M_{q}] is G_{q} = F_{q−2}. So, for example, the domain of C_{0} acting on the Bergman space A^{2} is the Hardy space H^{2} and the domain of [L_{0}, M_{0}] again acting on A^{2} is the Dirichlet space D.

The formulas are especially simple for the Hardy space H^{2} = F_{−1} in which all

the b_{k}(−1) = 1. An f ∈ H(𝔻) belongs to H^{2} if and only if ∑_{k=0}^{∞} |f_{k}|^{2} < ∞. Further,

C_{−1}z^{k} = \sqrt{k + 1} z^{k+1}, A_{−1}z^{k} = \sqrt{k} z^{k−1},

L_{−1}f(z) = f_{1} + ∑_{k=1}^{∞}(\sqrt{k} f_{k−1} + \sqrt{k + 1} f_{k+1})z^{k},

and M_{−1} is similar. The functions that give equality in the uncertainty relation for λ = μ = 0 are complex scalar multiples of

f^{(−1)}(z) = ∑_{l=0}^{∞}(\frac{γ + 1}{γ − 1})^{l}\sqrt{\frac{1 · 3 · 5 ⋯ (2l − 1)}{2 · 4 · 6 ⋯ 2l}} z^{2l}

with γ < 0.

5. Uncertainty Principles in Spaces on the Ball

We now let n > 1, use the full multivariable orthonormal basis B_{b} of F_{b} on 𝔹 given in (10), and define creation and annihilation operator tuples.

Definition 5.1. For j = 1, …, n, we define the operators C_{b_{j}} and A_{b_{j}} on F_{b} by

C_{b_{j}}e_{α}^{b} := \sqrt{α_{j} + 1} e_{α+ϵ_{j}}^{b}  and  A_{b_{j}}e_{α}^{b} := \sqrt{α_{j}} e_{α−ϵ_{j}}^{b} (α_{j} ≥ 1),

and by A_{b_{j}}e_{α}^{b} = 0 if α_{j} = 0. We also define the operator tuples C_{b} := (C_{b_{1}}, …, C_{b_{n}}) and A_{b} := (A_{b_{1}}, …, A_{b_{n}}).

The C_{b_{j}} and A_{b_{j}} can be called weighted shift operators, but they shift to basis elements that are not immediate neighbors. They are densely defined operators since they are defined at least on polynomials and they are unbounded. For example, C_{b_{j}} is unbounded at least on a subsequence of B_{b} starting with any e_{α}^{b} and goes by increasing α_{j} by 1 each time.

As before, A_{b_{j}} = C_{b_{j}}^{∗} and C_{b_{j}} = A_{b_{j}}^{∗} for j = 1, …, n, so they are all closed operators. Also

(26) C_{b_{j}}z^{α} = \sqrt{|α| + 1} \sqrt{\frac{b_{|α|+1}}{b_{|α|}}} z^{α+ϵ_{j}}  and  A_{b_{j}}z^{α} = \frac{α_{j}}{\sqrt{|α|}} \sqrt{\frac{b_{|α|−1}}{b_{|α|}}} z^{α−ϵ_{j}},

the latter for α_{j} ≥ 1. Hence

C_{b_{j}}A_{b_{j}}z^{α} = α_{j}z^{α}  and  A_{b_{j}}C_{b_{j}}z^{α} = (α_{j} + 1)z^{α}.

Thus

\begin{aligned}(C_{b} · A_{b})f(z) & = ∑_{|α|=1}^{∞} |α|f_{α}z^{α} = Nf(z), \\ (A_{b} · C_{b})f(z) & = ∑_{|α|=0}^{∞}(|α| + n)f_{α}z^{α} = Nf(z) + nIf(z), \end{aligned} ; (27)

where N is the number operator as before, resulting in

(A_{b} · C_{b} − C_{b} · A_{b})f = nf,  that is,  [A_{b}, C_{b}] = nI

on the intersection of the domains of A_{b} · C_{b} and C_{b} · A_{b}.

The domains of C_{b_{j}} and A_{b_{j}} are obtained using (26) and (9) as

D(C_{b_{j}}) = D(A_{b_{j}}) = \{ f ∈ H(𝔹) : ∑_{|α|=1}^{∞} \frac{1}{b_{|α|}} \frac{α!}{|α|!} α_{j} |f_{α}|^{2} < ∞ \}.

Then the domain of the tuples C_{b} and A_{b} is

E_{b} := \{ f ∈ H(𝔹) : ∑_{|α|=1}^{∞} \frac{1}{b_{|α|}} \frac{α!}{|α|!} |α| |f_{α}|^{2} < ∞ \}.

Similarly, the domain of A_{b} · C_{b} and C_{b} · A_{b} is

(28)  G_{b} := \{ f ∈ H(𝔹) : ∑_{|α|=1}^{∞} \frac{1}{b_{|α|}} \frac{α!}{|α|!} |α|^{2}|f_{α}|^{2} < ∞ \}.

We define further operator tuples by L_{b} := C_{b} + A_{b} = (C_{b_{1}} + A_{b_{1}}, …, C_{b_{n}} + A_{b_{n}}) and by M_{b} := i(C_{b} − A_{b}) similarly. Explicitly,

L_{b_{j}}f(z) = \frac{1}{\sqrt{b_{1}}} f_{ϵ_{j}} + ∑_{|α|=1}^{∞}(\sqrt{|α|} \sqrt{\frac{b_{|α|}}{b_{|α|−1}}} f_{α−ϵ_{j}} + \frac{α_{j} + 1}{\sqrt{|α| + 1}} \sqrt{\frac{b_{|α|}}{b_{|α|+1}}} f_{α+ϵ_{j}}) z^{α},

and M_{b_{j}}f(z) is similar. The C_{b_{j}} commute with each other and so do the A_{b_{j}}. Also C_{b_{j}} and A_{b_{k}} commute if j ≠ k. Then the L_{b_{j}} commute among themselves and so do the M_{b_{j}}. Further, D(L_{b}) = D(M_{b}) = E_{b}.

We call L_{b} and M_{b} self-adjoint due to L_{b_{j}}^{∗} = L_{b_{j}} and M_{b_{j}}^{∗} = M_{b_{j}} for each j = 1, …, n. Since the sums in L_{b} and M_{b} and the products in L_{b} · M_{b} and M_{b} · L_{b} are applied componentwise, we readily obtain

[L_{b}, M_{b}] = 2i[A_{b}, C_{b}] = 2inI

on G_{b} = D([L_{b}, M_{b}]). We then obtain the following theorem which introduces a joint average uncertainty inequality for operator tuples.

Theorem 5.2. For f ∈ G_{b} ⊂ F_{b} and λ_{j}, μ_{j} ∈ ℝ for j = 1, …, n,

\frac{1}{n} ∑_{j=1}^{n} ∥(C_{b_{j}} + A_{b_{j}} − λ_{j}I)f∥_{b} ∥(C_{b_{j}} − A_{b_{j}} − iμ_{j}I)f∥_{b} ≥ ∥f∥_{b}^{2}.

For n = 2 and λ_{j} = μ_{j} = 0 for j = 1, 2, equality holds for a function in G_{b} if and only if it is a linear combination of f^{b} and g^{b} given in (32) and (33) in the proof.

Proof. Using (5) with λ and μ in place of τ and σ, for f ∈ G_{b},

2in⟨f, f⟩_{b} = ⟨[L_{b}, M_{b}]f, f⟩_{b} = ⟨[L_{b} − λI, M_{b} − μI]f, f⟩_{b}

= ⟨(L_{b} − λI) · (M_{b} − μI)f, f⟩_{b} − ⟨(M_{b} − μI) · (L_{b} − λI)f, f⟩_{b}

= ∑_{j=1}^{n} (⟨M_{b_{j}}f − μ_{j}f, L_{b_{j}}f − λ_{j}f⟩_{b} − ⟨L_{b_{j}}f − λ_{j}f, M_{b_{j}}f − μ_{j}f⟩_{b})

= 2i ∑_{j=1}^{n} Im⟨M_{b_{j}}f − μ_{j}f, L_{b_{j}}f − λ_{j}f⟩_{b}.

Hence on G_{b},

∥f∥_{b}^{2} ≤ \frac{1}{n} ∑_{j=1}^{n} |⟨M_{b_{j}}f − μ_{j}f, L_{b_{j}}f − λ_{j}f⟩_{b}| ≤ \frac{1}{n} ∑_{j=1}^{n} ∥M_{b_{j}}f − μ_{j}f∥_{b}∥L_{b_{j}}f − λ_{j}f∥_{b}.

For equality, we only work out the case indicated in the statement of the theorem, simply because computations in the general case are too tedious. But we can let n > 1 be arbitrary in the initial steps. As in the proof of Theorem 4.3, equality holds in the first inequality if and only if each ⟨M_{b_{j}}f, L_{b_{j}}f⟩_{b} is pure imaginary with positive imaginary part. Equality holds in the second inequality if and only if L_{b_{j}}f = β_{j}M_{b_{j}}f for some β_{j} ∈ ℂ for each j. The two conditions together imply

that equality holds in the uncertainty inequality for an f^{b}(z) = ∑_{|α|=0}^{∞} f_{α}^{b}z^{α} ∈ G_{b} if

and only if L_{b_{j}}f^{b} = iγ_{j}M_{b_{j}}f^{b} for some γ_{j} < 0. Equivalently,

\begin{aligned}& \frac{1}{\sqrt{b_{1}}} f_{ϵ_{j}}^{b} + ∑_{|α|=1}^{∞}(\sqrt{|α|} \sqrt{\frac{b_{|α|}}{b_{|α|−1}}} f_{α−ϵ_{j}}^{b} + \frac{α_{j} + 1}{\sqrt{|α + 1}} \sqrt{\frac{b_{|α|}}{b_{|α|+1}}} f_{α+ϵ_{j}}^{b}) z^{α} \\ = & \frac{γ_{j}}{\sqrt{b_{1}}} f_{ϵ_{j}}^{b} + γ_{j} ∑_{|α|=1}^{∞}(\frac{α_{j} + 1}{\sqrt{|α + 1}} \sqrt{\frac{b_{|α|}}{b_{|α|+1}}} f_{α+ϵ_{j}}^{b} − \sqrt{|α|} \sqrt{\frac{b_{|α|}}{b_{|α|−1}}} f_{α−ϵ_{j}}^{b}) z^{α} \end{aligned} ; (29)

for each j.

For each j, the constant terms of (29) give (1 − γ_{j})f_{ϵ_{j}}^{b} = 0 which yields

(30)  f_{ϵ_{j}}^{b} = 0

since γ_{j} < 0. Setting the coefficients of z^{α} on both sides of (29) equal to each other and using γ_{j} < 0, we obtain

(31)  f_{α+ϵ_{j}}^{b} = \frac{γ_{j} + 1}{γ_{j} − 1} \frac{\sqrt{|α|} \sqrt{|α| + 1}}{α_{j} + 1} \sqrt{\frac{b_{|α|+1}}{b_{|α|−1}}} f_{α−ϵ_{j}}^{b}.

From now on, we consider only the case n = 2. So α = (α_{1}, α_{2}), j = 1, 2, and we have only ϵ_{1} = (1, 0) and ϵ_{2} = (0, 1). By (30) and (31), f_{(odd,even)}^{b} = 0 and f_{(even,odd)}^{b} = 0. Further by (31), every f_{(even,even)}^{b} depends on f_{(0,0)}^{b}. Similar to the proof of Theorem 4.3, with α = (2l, 2m), we obtain

f_{(2l,2m)}^{b} = (\frac{γ_{1} + 1}{γ_{1} − 1})^{l}(\frac{γ_{2} + 1}{γ_{2} − 1})^{m} \frac{\sqrt{(2l + 2m)!}}{2^{l+m} l! m!} \sqrt{b_{2l+2m}} f_{(0,0)}^{b}

for l, m = 0, 1, 2, . . .. Denoting the third factor by p_{lm}, these coefficients define

(32)  f^{b}(z_{1}, z_{2}) = ∑_{l,m=0}^{∞}(\frac{γ_{1} + 1}{γ_{1} − 1})^{l}(\frac{γ_{2} + 1}{γ_{2} − 1})^{m}p_{lm} \sqrt{b_{2l+2m}} z_{1}^{2l}z_{2}^{2m}.

Again by (31), every f_{(odd,odd)}^{b} depends on f_{(1,1)}^{b}. With α = (2l + 1, 2m + 1), similar to above, we obtain

f_{(2l+1,2m+1)}^{b} = (\frac{γ_{1}+1}{γ_{1}−1})^{l}(\frac{γ_{2}+1}{γ_{2}−1})^{m}\frac{\sqrt{(2l + 2m + 2)!}}{1·3 ⋯ (2l+1)·1·3 ⋯ (2m+1)} \sqrt{\frac{b_{2l+2m+2}}{2b_{2}}} f_{(1,1)}^{b}

for l, m = 0, 1, 2, . . .. Denoting the third factor by q_{lm}, these coefficients define

(33)  g^{b}(z_{1}, z_{2}) = ∑_{l,m=0}^{∞}(\frac{γ_{1} + 1}{γ_{1} − 1})^{l}(\frac{γ_{2} + 1}{γ_{2} − 1})^{m}q_{lm} \sqrt{\frac{b_{2l+2m+2}}{2b_{2}}} z_{1}^{2l+1}z_{2}^{2m+1}.

We must also check if f^{b} and g^{b} belong to G_{b}. It is a routine calculation that

\frac{α!}{|α|!} p_{lm}^{2} ≤ 1  and  \frac{α!}{|α|!} q_{lm}^{2} ≤ 1.

Using the first, it is straightforward to see that the sum in (28) is

≤ 8 ∑_{l,m=1}^{∞}(\frac{γ_{1} + 1}{γ_{1} − 1})^{2l}(\frac{γ_{2} + 1}{γ_{2} − 1})^{2m}(l^{2} + m^{2}),

which is finite if and only if γ_{1} < 0 and γ_{2} < 0. This shows f^{b} ∈ G_{b} for all γ_{1} < 0 and γ_{2} < 0. Similarly, g^{b} ∈ G_{b} for all γ_{1} < 0 and γ_{2} < 0. □

Specializing to F_{q}, by Definition 3.3,

L_{q_{j}}f(z) = \frac{f_{ϵ_{j}}}{\sqrt{1 + n + q}} + ∑_{|α|=1}^{∞}(\sqrt{n + q + |α|} f_{α−ϵ_{j}} + \frac{(1 + α_{j})f_{α+ϵ_{j}}}{\sqrt{1 + n + q + |α|}})z^{α}

if q > −(1 + n) and it equals

\sqrt{1 − n − q} f_{ϵ_{j}} + ∑_{|α|=1}^{∞}(\frac{|α| f_{α−ϵ_{j}}}{\sqrt{−n − q + |α|}} + \sqrt{1 − n − q + |α|} \frac{1 + α_{j}}{\sqrt{1 + |α|}} f_{α+ϵ_{j}})z^{α}

if q ≤ −(1 + n). The expressions for C_{q_{j}}f(z), A_{q_{j}}f(z), and M_{q_{j}}f(z) are similar. By (11), domains of operators behave as in n = 1: E_{q} = F_{q−1} and G_{q} = F_{q−2}.

For n > 1, there are four importamt values of q. Still q = 0 and q = −1 give the Bergman and Hardy spaces A^{2} and H^{2}. But q = −n gives the Drury-Arveson space A and q = −(1 + n) gives the Dirichlet space D. The formulas are simplest for A since the Drury-Arveson space is special for multivariable operator theory due to b_{k}(−n) = 1 for all k = 0, 1, 2, . . .. So

L_{(−n)_{j}}f(z) = f_{ϵ_{j}} + ∑_{|α|=1}^{∞}(\sqrt{|α|} f_{α−ϵ_{j}} + \frac{1 + α_{j}}{\sqrt{1 + |α|}} f_{α+ϵ_{j}})z^{α},

and with q = −n = −2,

f^{(−2)}(z_{1}, z_{2}) = ∑_{l,m=0}^{∞}(\frac{γ_{1} + 1}{γ_{1} − 1})^{l}(\frac{γ_{2} + 1}{γ_{2} − 1})^{m}p_{lm} z_{1}^{2l}z_{2}^{2m},

which have almost the same forms as the corresponding quantities in the Hardy space when n = 1.

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Bi̇lkent Üni̇versi̇tesi̇, Matemati̇k Bölümü, 06800 Ankara, Türki̇ye

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