Uncertainty principles in holomorphic function spaces on the unit ball

Abstract

On all Bergman–Besov Hilbert spaces on the unit disk, 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 $\langle \cdot \,,\cdot \rangle _H$ and the associated norm $\|\cdot \|_H$ . Then

(1) $$ \begin{align} \bigl\|(L-\lambda I)u\bigr\|_H\bigl\|(M-\mu I)u\bigr\|_H \ge\frac12\Bigl|\bigl\langle(LM-ML)u,u\bigr\rangle_H\Bigr| \end{align} $$

for all $\lambda ,\mu \in \mathbb R$ and all u that lie in the domain of both $LM$ and $ML$ . Equality holds if and only if $(L-\lambda I)u=i\gamma (M-\mu I)u$ for some $\gamma \in \mathbb R$ .

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, G], coincidentally on pages 27 and 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 e and $x,y\in Z$ , then $xy-yx\ne e$ .

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.

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 [So1, So2] on certain, but not all, Bergman and Dirichlet spaces on the unit disk, but in these sources, one of the operators is always taken as the first-order derivative, resulting in $[L,M]\ne cI$ . It is shown in [CD, Theorem 11] that there are no first-order self-adjoint differential operators on weighted Bergman spaces on the unit disk 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 disk 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 $\mathbb C^n$ which includes the Drury-Arveson space. The formulation with tuples of operators seems new.

Our main results are Theorems 4.3 and 5.2.

2 Notation and preliminaries

Let $\mathbb B$ be the open unit ball in $\mathbb C^n$ with respect to the usual Hermitian inner product $\langle z,w\rangle =z_1\overline w_1+\cdots +z_n\overline w_n$ , where we conjugate the second variable following the tradition in mathematics, and the associated norm $|z|=\sqrt {\langle z,z\rangle }$ . When $n=1$ , the ball is the unit disk $\mathbb D$ in the complex plane.

In multi-index notation, $\alpha =(\alpha _1,\ldots ,\alpha _n)$ is an n-tuple of nonnegative integers, $|\alpha |=\alpha _1+\cdots +\alpha _n$ , $\alpha !=\alpha _1!\cdots \alpha _n!$ , $0^0=1$ , and $z^{\alpha }=z_1^{\alpha _1}\cdots z_n^{\alpha _n}$ . We also let $\epsilon _j:=(0,\ldots ,0,1,0,\ldots ,0)$ , where $1$ is in the jth position and the right side of $:=$ defines its left side.

The Pochhammer symbol $(p)_q$ is defined by

$$ \begin{align*}(p)_q=\frac{\Gamma(p+q)}{\Gamma(p)}\end{align*} $$

when p and $p+q$ are off the pole set $-\mathbb N$ of the gamma function $\Gamma $ . This is a shifted rising factorial since $(p)_k=p(p+1)\cdots (p+k-1)$ for positive integer k. In particular, $(1)_k=k!$ and $(p)_0=1$ . Stirling formula gives

(2) $$ \begin{align} \frac{\Gamma(r+p)}{\Gamma(r+q)}\sim r^{p-q},\qquad \frac{(p)_r}{(q)_r}\sim r^{p-q},\qquad \frac{(r)_p}{(r)_q}\sim r^{p-q}\qquad(\operatorname{Re}r\to\infty), \end{align} $$

where $P\sim Q$ means that $|P/Q|$ is bounded above and below by two strictly positive constants, that is, $P=\mathcal O(Q)$ and $Q=\mathcal O(P)$ , 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 $\mathbb B$ if $K(z,\cdot )\in H$ for each $z\in \mathbb B$ and

$$ \begin{align*}u(z)=\langle u(\cdot),K(z,\cdot)\rangle_H\qquad(u\in H,\;z\in\mathbb B).\end{align*} $$

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 $H(\mathbb B)$ .

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], $C\subset C^{**}=A^*$ , 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) $$ \begin{align} [L,M]=2i[A,C]. \end{align} $$

Moreover, if $\lambda ,\mu \in \mathbb R$ , then $L-\lambda I$ and $M-\mu I$ are also self-adjoint and

(4) $$ \begin{align} [L-\lambda I,M-\mu I]=[L,M]. \end{align} $$

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 $[A,C]=I$ .

1. (i) $\|(C+A-\lambda I)u\|_H\|(C-A-i\mu I)u\|_H\ge \|u\|_H^2$ for all $\lambda ,\mu \in \mathbb R$ and all u that lie in the domain of both $CA$ and $AC$ .

2. (ii) $\|(C+A)u\|_H^2+\|(C-A)u\|_H^2\ge 2\|u\|_H^2$ for all u that lie in the domain of both $CA$ and $AC$ .

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

If $T=(T_1,\ldots ,T_n)$ and $S=(S_1,\ldots ,S_n)$ are tuples of operators on the same Hilbert space H, we use the notation $T\cdot S:=T_1S_1+\cdots +T_nS_n$ . We define the commutator of the tuples T and S by $[T,S]:=T\cdot S-S\cdot T$ . With these definitions, if $\tau I=(\tau _1I,\ldots ,\tau _nI)$ with $\tau _j\in \mathbb C$ , $j=1,\ldots ,n$ , and $\sigma I$ is similar, then by a straightforward calculation,

(5) $$ \begin{align} [T-\tau I,S-\sigma I]=[T,S]. \end{align} $$

3 Weighted symmetric Fock spaces

In [Ka], large families of weighted symmetric (bosonic) Fock spaces of holomorphic functions on $\mathbb B$ 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,\ldots $ , and

(6) $$ \begin{align} \limsup_{k\to\infty}\,b_k^{1/k}\le1. \end{align} $$

We define positive-definite kernels by

(7) $$ \begin{align} K_b(z,w):=\sum_{k=0}^\infty b_k\langle z,w\rangle^k =\sum_{k=0}^\infty b_k\sum_{|\alpha|=k}\frac{|\alpha|!}{\alpha!} z^{\alpha}\overline w^\alpha\qquad(z,w\in\mathbb B) \end{align} $$

and spaces $\mathcal F_b$ 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 $\mathbb B\times \mathbb B$ , thereby defining $K_b$ as a holomorphic function of $z\in \mathbb B$ and a conjugate holomorphic function of $w\in \mathbb B$ .

Theorem 3.2. The space $\mathcal F_b$ consists of all $f\in H(\mathbb B)$ with Taylor expansions

(8) $$ \begin{align} f(z)=\sum_{|\alpha|=0}^\infty f_\alpha z^{\alpha} \end{align} $$

converging absolutely and uniformly on compact subsets of $\mathbb B$ for which

(9) $$ \begin{align} \|f\|_b^2 :=\sum_{|\alpha|=0}^\infty\frac1{b_{|\alpha|}}\frac{\alpha!}{|\alpha|!} |f_\alpha|^2<\infty \end{align} $$

and is equipped with the inner product

$$ \begin{align*}\langle f,g\rangle_b :=\sum_{|\alpha|=0}^\infty\frac1{b_{|\alpha|}}\frac{\alpha!}{|\alpha|!} f_\alpha\overline g_\alpha.\end{align*} $$

Further,

(10) $$ \begin{align} \mathcal B_b:=\biggl\{\,e_\alpha^b(z) :=\sqrt{b_{|\alpha|}\frac{|\alpha|!}{\alpha!}}\,z^{\alpha} :\alpha\in\mathbb N^n\,\biggr\} \end{align} $$

is an orthonormal basis for $\mathcal F_b$ . Moreover, holomorphic polynomials in the n variables $z_1,\ldots ,z_n$ are dense in each $\mathcal F_b$ .

In particular, for each $\alpha \in \mathbb N^n$ ,

$$ \begin{align*}\|z^{\alpha}\|_b^2=\frac1{b_{|\alpha|}} \frac{\alpha!}{|\alpha|!}.\end{align*} $$

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 $\mathbb B$ .

Definition 3.3. For $q\in \mathbb R$ and $k=0,1,2,\ldots $ , we set

$$ \begin{align*}b_k(q) :=\begin{cases} \dfrac{(1+n+q)_k}{k!},&\text{if }q>-(1+n),\\ \dfrac{k!}{(1-(n+q))_k},&\text{if }q\le-(1+n), \end{cases}\end{align*} $$

denote by $K_q(z,w):=\sum \limits _{k=0}^\infty b_k(q)\langle z,w\rangle ^k$ the reproducing kernel with coefficient sequence $(b_k(q))_k$ and by $\mathcal F_q$ the Hilbert space generated by the kernel $K_q$ .

By (2),

(11) $$ \begin{align} b_k(q)\sim k^{n+q}\qquad(k\to\infty) \end{align} $$

for any $q\in \mathbb R$ assuring that (6) is satisfied. So an $f\in H(\mathbb B)$ given by (8) belongs to $\mathcal F_q$ if and only if

(12) $$ \begin{align} \sum_{|\alpha|=1}^\infty\frac1{|\alpha|^{n+q}}\,\frac{\alpha!}{|\alpha|!} \,|f_\alpha|^2<\infty. \end{align} $$

Note that

$$ \begin{align*} K_q(z,w) =\begin{cases} \displaystyle\frac1{(1-\langle{z,w}\rangle)^{1+n+q}}={}_{2}{F}_{1}(1+n+q,1;1;\langle{z,w}\rangle),&\text{if }q>-(1+n),\\ {}_{2}{F}_{1}(1,1;1-(n+q);\langle{z,w}\rangle),&\text{if }q\le-(1+n), \end{cases}\end{align*} $$

where ${}_{2}{F}_1$ is the Gauss hypergeometric function. In particular,

$$ \begin{align*}K_{-(1+n)}(z,w)=\frac1{\langle z,w\rangle} \,\log\frac1{1-\langle z,w\rangle}.\end{align*} $$

Thus $\mathcal F_q$ is the weighted Bergman space $A_q^2$ for $q>-1$ , the Hardy space $H^2$ for $q=-1$ , the Drury–Arveson space $\mathcal A$ for $q=-n$ , and the Dirichlet space $\mathcal D$ 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 $\mathcal F_q$ are bounded on $\mathbb B$ . The inner products and hence the norms of all the spaces in the $\mathcal F_q$ family can be expressed as integrals on $\mathbb B$ of either the functions or their sufficiently high-order derivatives (see [Ka] for details).

4 Uncertainty principles in spaces on the disk

We start with the case of the function spaces on the unit disk, that is, $n=1$ . Many of the formulas in Section 3 are simplified mainly because now $|\alpha |=\alpha =k$ . So the terms of the orthonormal basis $\mathcal B_b$ of $\mathcal F_b$ are $e_k^b(z)=\sqrt {b_k}\,z^k$ for $k=0,1,2,\ldots $ ; in particular, $e_0^b=1$ . Equivalently,

(13) $$ \begin{align} \|z^k\|_b^2=\frac1{b_k}\qquad(k=0,1,2,\ldots). \end{align} $$

The homogeneous expansion of a function $f\in H(\mathbb D)$ is now its Taylor expansion which can also be written in terms of the orthonormal basis of $\mathcal F_b$ as

(14) $$ \begin{align} f(z)=\sum_{k=0}^\infty f_kz^k =\sum_{k=0}^\infty\frac{f_k}{\sqrt{b_k}}\,e_k^b(z)\qquad(z\in\mathbb D). \end{align} $$

Definition 4.1. We define the operator $C_b$ on $\mathcal F_b$ by first letting

$$ \begin{align*}C_be_k^b:=\sqrt{k+1}\,e_{k+1}^b\end{align*} $$

on $\mathcal B_b$ and then by extending it linearly to its span.

Thus, the operator $C_b$ on $\mathcal F_b$ 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 $\mathcal F_b$ . However, it is densely defined since polynomials are dense in every $\mathcal F_b$ .

By [Sh] or in a similar way to the unweighted shifts on $\mathcal F_b$ studied in [Ka], the adjoint $A_b:=C_b^*$ of $C_b$ is given by

$$ \begin{align*}A_be_k^b=\sqrt{k}\,e_{k-1}^b\qquad(k\ge1)\end{align*} $$

and $A_be_0^b=0$ . Similarly, $A_b$ is densely defined and unbounded on $\mathcal F_b$ , 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 $f\in \mathcal F_b$ is given by (14), then

(15) $$ \begin{align} C_bf(z)&=\sum_{k=0}^\infty\sqrt{k+1}\,\sqrt{\frac{b_{k+1}}{b_k}}f_kz^{k+1},\\ A_bf(z)&=\sum_{k=1}^\infty\sqrt{k}\,\sqrt{\frac{b_{k-1}}{b_k}}f_kz^{k-1}.\nonumber \end{align} $$

Continuing,

$$ \begin{align*}A_bC_be_k^b=(k+1)e_k^b\qquad\text{and}\qquad C_bA_be_k^b=ke_k^b\quad (k\ge1).\end{align*} $$

If $f\in \mathcal F_b$ , then

(16) $$ \begin{align} C_bA_bf(z)&=\sum_{k=1}^\infty kf_kz^k=:Nf(z), \\ A_bC_bf(z)&=\sum_{k=0}^\infty(k+1)f_kz^k=Nf(z)+If(z),\nonumber \end{align} $$

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

(17) $$ \begin{align} (A_bC_b-C_bA_b)f=f,\qquad\text{that is,}\qquad[A_b,C_b]=I \end{align} $$

on the subspace of $\mathcal F_b$ that is the intersection of the domains if $A_bC_b$ and $C_bA_b$ . 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 us describe the domains of the operators. The domain $D(C_b)$ of $C_b$ is the subspace of all $f\in \mathcal F_b$ for which also $C_bf\in \mathcal F_b$ . So, by Theorem 3.2, (13), (15), and (16),

(18) $$ \begin{align} \mathcal E_b&:=\biggl\{\,f\in H(\mathbb D):\sum_{k=1}^\infty \,\frac k{b_k}\,|f_k|^2<{\infty}\,\biggr\}=D(C_b)=D(A_b),\\ \mathcal G_b&:=\biggl\{\,f\in H(\mathbb D):\sum_{k=1}^\infty \,\frac{k^2}{b_k}\,|f_k|^2<{\infty}\,\biggr\}=D(A_bC_b)=D(C_bA_b). \nonumber \end{align} $$

It is clear that $\mathcal G_b\subset \mathcal E_b\subset \mathcal F_b$ and each inclusion is dense since all three sets contain polynomials. By [R, Theorem 13.9], we conclude the following.

Proposition 4.2. $C_b$ , $A_b$ , $A_bC_b$ , and $C_bA_b$ 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

$$ \begin{align*} L_bf(z)&=\sqrt{b_1}\,f_0z+\sum_{k=1}^\infty\biggl(\sqrt{k+1} \,\sqrt{\frac{b_{k+1}}{b_k}}\,z^2+\sqrt{k}\,\sqrt{\frac{b_{k-1}}{b_k}}\biggr) f_kz^{k-1}\\ &=\frac1{\sqrt{b_1}}\,f_1+\sum_{k=1}^\infty\Biggl(\sqrt{k}\, \sqrt{\frac{b_k}{b_{k-1}}}\,f_{k-1}+\sqrt{k+1}\,\sqrt{\frac{b_k}{b_{k+1}}} \,f_{k+1}\Biggr)\,{z}^{k}, \end{align*} $$

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 $M_bf(z)$ since it is so similar to $L_bf(z)$ . Both $L_b$ and $M_b$ are closed operators with domain $\mathcal E_b$ . By (3) and (17),

(19) $$ \begin{align} [L_b,M_b]=2iI \end{align} $$

on $\mathcal G_b=D([L_b,M_b])$ . The uncertainty principle in $\mathcal F_b$ implied by Corollary 2.2(i) is the following.

Theorem 4.3. For $f\in \mathcal G_b\subset \mathcal F_b$ and $\lambda ,\mu \in \mathbb R$ ,

$$ \begin{align*}\|(C_b+A_b-\lambda I)f\|_b\,\|(C_b-A_b-i\mu I)f\|_b\ge\|f\|_b^2.\end{align*} $$

For $\lambda =\mu =0$ , equality holds for a function in $\mathcal G_b$ if and only if it is a complex scalar multiple of

$$ \begin{align*}f^b(z)=\sum_{l=0}^\infty\biggl(\frac{\gamma+1}{\gamma-1}\biggr)^l \sqrt{\frac{1\cdot3\cdot5\cdots(2l-1)}{2\cdot4\cdot6\cdots2l}} \,\sqrt{b_{2l}}\;z^{2l}\end{align*} $$

for some $\gamma <0$ . For $(\lambda ,\mu )\ne (0,0)$ , Taylor series coefficients of the functions that give equality can be obtained from a three-term recurrence relation.

Proof By (19) and (4), for $\lambda ,\mu \in \mathbb R$ ,

$$ \begin{align*} 2i\langle f,f\rangle_b &=\bigl\langle[L_b,M_b]f,f\bigr\rangle_b =\bigl\langle[L_b-\lambda I,M_b-\mu I]f,f\bigr\rangle_b\\ &=\bigl\langle(L_b-\lambda I)(M_b-\mu I)f,f\bigr\rangle_b -\bigl\langle(M_b-\mu I)(L_b-\lambda I)f,f\bigr\rangle_b\\ &=\bigl\langle(M_b-\mu I)f,(L_b-\lambda I)f\bigr\rangle_b -\overline{\bigl\langle(M_b-\mu I)f,(L_b -\lambda I)f\bigr\rangle_b}\\ &=2i\operatorname{Im}\bigl\langle(M_b-\mu I)f, (L_b-\lambda I)f\bigr\rangle_b. \end{align*} $$

Hence,

$$ \begin{align*}\|f\|_b \le\bigl|\bigl\langle(M_b-\mu I)f,(L_b-\lambda I)f\bigr\rangle_b\bigr| \le\|(M_b-\mu I)f\|_b\|(L_b-\lambda I)f\|_b\end{align*} $$

on $\mathcal G_b$ , which is the desired uncertainty inequality.

Equality holds in the first inequality if and only if $\langle M_bf,L_bf\rangle _b$ is pure imaginary with positive imaginary part. Equality holds in the second inequality if and only if $L_bf=\beta M_bf$ for some $\beta \in \mathbb C$ . Applying the two conditions together, we deduce that equality holds in the uncertainty inequality for an $f^b(z)=\sum \limits _{k=0}^\infty f_k^bz^k\in \mathcal G_b$ if and only if $L_bf^b-\lambda f^b=i\gamma (M_bf^b-\mu f^b)$ for some $\gamma <0$ . Equivalently,

(20) $$ \begin{align} \frac{f_1^b}{\sqrt{b_1}}&-\lambda f_0^b +\sum_{k=1}^\infty\Biggl(\sqrt{k}\,\sqrt{\frac{b_k}{b_{k-1}}}\,f_{k-1}^b +\sqrt{k\!+\!1}\,\sqrt{\frac{b_k}{b_{k+1}}}\,f_{k+1}^b-\lambda f_k^b\Biggr) \,z^k \\ =&\frac{\gamma f_1^b}{\sqrt{b_1}}-i\gamma\mu f_0^b+\gamma\sum_{k=1}^\infty \Biggl(\sqrt{k\!+\!1}\,\sqrt{\frac{b_k}{b_{k\!+\!1}}}\,f_{k\!+\!1}^b -\sqrt{k}\,\sqrt{\frac{b_k}{b_{k\!-\!1}}}\,f_{k\!-\!1}^b-i\mu f_k^b\Biggr) \,z^k.\nonumber \end{align} $$

Setting the coefficients of $z^k$ on both sides of (20) equal to each other, letting $f^b_{-1}=0$ for convenience, and using $\gamma <0$ , we obtain

(21) $$ \begin{align} f_{k+1}^b=\frac{\gamma+1}{\gamma-1}\,\sqrt{\frac k{k+1}} \,\sqrt{\frac{b_{k+1}}{b_{k-1}}}\;f_{k-1}^b -\frac{\lambda-i\gamma\mu}{\gamma-1}\,\frac1{\sqrt{k+1}} \,\sqrt{\frac{b_{k+1}}{b_k}}\;f_k^b \end{align} $$

for $k=0,1,2,\ldots $ . 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 $\gamma <0$ this $f^b$ belongs to $\mathcal G_b$ . But the computations are cumbersome and we only work out the details of the representative case $\lambda =\mu =0$ . As a side note, if $\gamma =1$ , then the only function satisfying (20) is the zero function.

For $\lambda =\mu =0$ , the constant terms of (20) give

(22) $$ \begin{align} (1-\gamma)f_1^b=0 \end{align} $$

and the recurrence relation (21) takes the form

(23) $$ \begin{align} f_{k+1}^b=\frac{\gamma+1}{\gamma-1}\,\sqrt{\frac k{k+1}} \,\sqrt{\frac{b_{k+1}}{b_{k-1}}}\;f_{k-1}^b. \end{align} $$

Since $\gamma <0$ , (22) gives $f_1^b=0$ . Then, from (23), $f_3^b=f_5^b=\cdots =0$ as well. Letting now $k=2l-1$ , the coefficients with even indices satisfy

$$ \begin{align*}f_{2l}^b=\biggl(\frac{\gamma+1}{\gamma-1}\biggr)^l \sqrt{\frac{1\cdot3\cdot5\cdots(2l-1)}{2\cdot4\cdot6\cdots2l}} \,\sqrt{b_{2l}}\;f_0^b\qquad(l=1,2,\ldots).\end{align*} $$

Denoting the first square root by $d_{2l}$ and setting $d_0=1$ , we have

(24) $$ \begin{align} f^b(z)=f_0^b\sum_{l=0}^\infty\biggl(\frac{\gamma+1}{\gamma-1}\biggr)^l d_{2l}\,\sqrt{b_{2l}}\;z^{2l}=f_0^b\sum_{l=0}^\infty c_{2l}^bz^{2l}. \end{align} $$

Simply $d_{2l}\le 1/\sqrt 2$ and by (18), $f^b\in \mathcal G_b$ if and only if

(25) $$ \begin{align} \sum_{l=1}^\infty\frac{l^2}{b_{2l}}|c_{2l}^b|^2 =\sum_{l=1}^\infty\,l^2\biggl|\frac{\gamma+1}{\gamma-1}\biggr|^{2l}d_{2l}^2 <\infty. \end{align} $$

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

$$ \begin{align*}\sum_{l=1}^\infty l^2\biggl|\frac{\gamma+1}{\gamma-1}\biggr|^{2l},\end{align*} $$

and this holds when $|\gamma +1|<|\gamma -1|$ , that is, $\gamma <0$ .

Thus, the precise range of values of $\gamma $ for having a nontrivial $f^b\in \mathcal G_b$ given in (24) for which equality holds in the uncertainty inequality for $\lambda =\mu =0$ is $\gamma <0$ .

We next specialize to the $\mathcal F_q$ spaces of Definition 3.3 in which the $b_k(q)$ have specific values for $q\in \mathbb R$ , and use the subscript q as in $C_q$ to denote an operator on $\mathcal F_q$ . We have

$$ \begin{align*}C_qz^k=c_{k+1}^qz^{k+1} =\begin{cases} \sqrt{2+q+k}\;z^{k+1},&\text{if }q>-2,\\ \dfrac{k+1}{\sqrt{-q+k}}\,z^{k+1},&\text{if }q\le-2, \end{cases}\end{align*} $$

and

$$ \begin{align*}A_qz^k=a_{k-1}^qz^{k-1} =\begin{cases} \dfrac k{\sqrt{1+q+k}}\,z^{k-1} =\dfrac1{\sqrt{1+q+k}}\,(z^k)',&\text{if }q>-2,\\ \sqrt{-q+k-1}\;z^{k-1} =\dfrac{\sqrt{-q+k-1}}k\,(z^k)',&\text{if }q\le-2, \end{cases}\end{align*} $$

where primes denote differentiation. These explicit formulas show that both $C_q$ and $A_q$ are fractional differential operators of order $1/2$ since $c_k^q\sim a_k^q\sim k^{1/2}$ for each $q\in \mathbb R$ . For comparison, by (16), $C_bA_b$ and $A_bC_b$ are differential operators of order $1$ .

Three values of q give three important spaces; $\mathcal F_0$ is the Bergman space $A^2$ , $\mathcal F_{-1}$ is the Hardy space $H^2$ , and $\mathcal F_{-2}$ is the Dirichlet space $\mathcal D$ . 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 $f\in \mathcal F_q$ ,

$$ \begin{align*} L_qf(z)=\begin{cases}\displaystyle\frac1{\sqrt{2+q}}\,f_1+\sum_{k=1}^\infty \biggl(\sqrt{1+q+k}\;f_{k-1}+\frac{k+1}{\sqrt{2+q+k}}\,f_{k+1}\biggr)z^k, & \! \text{if }q>-2,\\ \displaystyle\sqrt{-q}\;f_1+\sum_{k=1}^\infty \biggl(\frac k{\sqrt{-1-q+k}}\,f_{k-1}+\sqrt{-q+k}\;f_{k+1}\biggr)z^k, & \! \text{if }q\le-2,\end{cases} \end{align*} $$

and $M_qf(z)$ is similar. So $L_q$ and $M_q$ are also fractional differential operators of order $1/2$ for each $q\in \mathbb R$ .

The domains of $C_q$ , $A_q$ , $L_q$ , and $M_q$ are all the same, $\mathcal E_q$ . A comparison of (18), (12), and (11) reveals that $\mathcal E_q=\mathcal F_{q-1}$ . Similarly, the domain of $A_qC_q$ , $C_qA_q$ , and $[L_q,M_q]$ is $\mathcal G_q=\mathcal F_{q-2}$ . 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 $\mathcal D$ .

The formulas are especially simple for the Hardy space $H^2=\mathcal F_{-1}$ in which all the $b_k(-1)=1$ . An $f\in H(\mathbb D)$ belongs to $H^2$ if and only if $\sum \limits _{k=0}^\infty |f_k|^2<\infty $ . Further, $C_{-1}z^k=\sqrt {k+1}\,z^{k+1}$ , $A_{-1}z^k=\sqrt {k}\,z^{k-1}$ ,

$$ \begin{align*}L_{-1}f(z) =f_1+\sum_{k=1}^\infty\bigl(\sqrt{k}\;f_{k-1}+\sqrt{k+1}\,f_{k+1}\bigr)z^k,\end{align*} $$

and $M_{-1}$ is similar. The functions that give equality in the uncertainty relation for $\lambda =\mu =0$ are complex scalar multiples of

$$ \begin{align*}f^{(-1)}(z)=\sum_{l=0}^\infty \biggl(\frac{\gamma+1}{\gamma-1}\biggr)^l \sqrt{\frac{1\cdot3\cdot5\cdots(2l-1)}{2\cdot4\cdot6\cdots2l}}\;z^{2l}\end{align*} $$

with $\gamma <0$ .

5 Uncertainty principles in spaces on the ball

We now let $n>1$ , use the full multivariable orthonormal basis $\mathcal B_b$ of $\mathcal F_b$ on $\mathbb B$ given in (10), and define creation and annihilation operator tuples.

Definition 5.1. For $j=1,\ldots ,n$ , we define the operators $C_{b_j}$ and $A_{b_j}$ on $\mathcal F_b$ by

$$ \begin{align*}C_{b_j}e_\alpha^b:=\sqrt{\alpha_j+1}\,e_{\alpha+\epsilon_j}^b \qquad\text{and}\qquad A_{b_j}e_\alpha^b:=\sqrt{\alpha_j}\,e_{\alpha-\epsilon_j}^b \quad(\alpha_j\ge1),\end{align*} $$

and by $A_{b_j}e_\alpha ^b=0$ if $\alpha _j=0$ . We also define the operator tuples $C_b:=(C_{b_1},\ldots ,C_{b_n})$ and $A_b:=(A_{b_1},\ldots ,A_{b_n})$ .

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 $\mathcal B_b$ starting with any $e_\alpha ^b$ and goes by increasing $\alpha _j$ by $1$ each time.

As before, $A_{b_j}=C_{b_j}^*$ and $C_{b_j}=A_{b_j}^*$ for $j=1,\ldots ,n$ , so they are all closed operators. Also,

(26) $$ \begin{align} C_{b_j}z^{\alpha}=\sqrt{|\alpha|+1} \,\sqrt{\frac{b_{|\alpha|+1}}{b_{|\alpha|}}}\,z^{\alpha+\epsilon_j} \qquad\text{and}\qquad A_{b_j}z^{\alpha}=\frac{\alpha_j}{\sqrt{|\alpha|}} \,\sqrt{\frac{b_{|\alpha|-1}}{b_{|\alpha|}}}\,z^{\alpha-\epsilon_j}, \end{align} $$

the latter for $\alpha _j\ge 1$ . Hence,

$$ \begin{align*}C_{b_j}A_{b_j}z^{\alpha}=\alpha_jz^{\alpha}\qquad\text{and}\qquad A_{b_j}C_{b_j}z^{\alpha}=(\alpha_j+1)z^{\alpha}.\end{align*} $$

Thus,

(27) $$ \begin{align} (C_b\cdot A_b)f(z)&=\sum_{|\alpha|=1}^\infty|\alpha|f_\alpha z^{\alpha} =Nf(z), \\ (A_b\cdot C_b)f(z)&=\sum_{|\alpha|=0}^\infty(|\alpha|+n)f_\alpha z^{\alpha} =Nf(z)+nIf(z),\nonumber \end{align} $$

where N is the number operator as before, resulting in

$$ \begin{align*}(A_b\cdot C_b-C_b\cdot A_b)f=nf,\qquad\text{that is,}\qquad[A_b,C_b]=nI\end{align*} $$

on the intersection of the domains of $A_b\cdot C_b$ and $C_b\cdot A_b$ .

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

$$ \begin{align*}D(C_{b_j})=D(A_{b_j})=\biggl\{\,f\in H(\mathbb B):\sum_{|\alpha|=1}^\infty \frac1{b_{|\alpha|}}\,\frac{\alpha!}{|\alpha|!}\,\alpha_j\,|f_\alpha|^2 <\infty\,\biggr\}.\end{align*} $$

Then the domain of the tuples $C_b$ and $A_b$ is

$$ \begin{align*}\mathcal E_b:=\biggl\{\,f\in H(\mathbb B): \sum_{|\alpha|=1}^\infty \frac1{b_{|\alpha|}}\,\frac{\alpha!}{|\alpha|!} \,|\alpha|\,|f_\alpha|^2<\infty\,\biggr\}.\end{align*} $$

Similarly, the domain of $A_b\cdot C_b$ and $C_b\cdot A_b$ is

(28) $$ \begin{align} \mathcal G_b:=\biggl\{\,f\in H(\mathbb B): \sum_{|\alpha|=1}^\infty \frac1{b_{|\alpha|}}\,\frac{\alpha!}{|\alpha|!} \,|\alpha|^2|f_\alpha|^2<\infty\,\biggr\}. \end{align} $$

We define further operator tuples by $L_b:=C_b+A_b=(C_{b_1}+A_{b_1},\ldots ,C_{b_n}+A_{b_n})$ and by $M_b:=i(C_b-A_b)$ similarly. Explicitly,

$$ \begin{align*}L_{b_j}f(z)=\frac1{\sqrt{b_1}}\,f_{\epsilon_j}+\sum_{|\alpha|=1}^\infty \Biggl(\sqrt{|\alpha|}\,\sqrt{\frac{b_{|\alpha|}}{b_{|\alpha|-1}}} \,f_{\alpha-\epsilon_j}+\frac{\alpha_j+1}{\sqrt{|\alpha|+1}} \,\sqrt{\frac{b_{|\alpha|}}{b_{|\alpha|+1}}}\,f_{\alpha+\epsilon_j}\Biggr) \,z^{\alpha},\end{align*} $$

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 $j\ne k$ . Then the $L_{b_j}$ commute among themselves and so do the $M_{b_j}$ . Further, $D(L_b)=D(M_b)=\mathcal E_b$ .

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 $j=1,\ldots ,n$ . Since the sums in $L_b$ and $M_b$ and the products in $L_b\cdot M_b$ and $M_b\cdot L_b$ are applied componentwise, we readily obtain

$$ \begin{align*}[L_b,M_b]=2i[A_b,C_b]=2inI\end{align*} $$

on $\mathcal G_b=D([L_b,M_b])$ . We then obtain the following theorem, which introduces a joint average uncertainty inequality for operator tuples.

Theorem 5.2. For $f\in \mathcal G_b\subset \mathcal F_b$ and $\lambda _j,\mu _j\in \mathbb R$ , for $j=1,\ldots ,n$ ,

$$ \begin{align*}\frac1n\sum_{j=1}^n\, \|(C_{b_j}+A_{b_j}-\lambda_jI)f\|_b\,\|(C_{b_j}-A_{b_j}-i\mu_jI)f\|_b \ge\|f\|_b^2.\end{align*} $$

For $n=2$ and $\lambda _j=\mu _j=0$ , for $j=1,2$ , equality holds for a function in $\mathcal G_b$ 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 $\lambda $ and $\mu $ in place of $\tau $ and $\sigma $ , for $f\in \mathcal G_b$ ,

$$ \begin{align*} 2in\langle f,f\rangle_b &=\bigl\langle[L_b,M_b]f,f\bigr\rangle_b =\bigl\langle[L_b-\lambda I,M_b-\mu I]f,f\bigr\rangle_b\\ &=\bigl\langle(L_b-\lambda I)\cdot(M_b-\mu I)f,f\bigr\rangle_b -\bigl\langle(M_b-\mu I)\cdot(L_b-\lambda I)f,f\bigr\rangle_b\\ &=\sum_{j=1}^n\,\bigl(\langle M_{b_j}f-\mu_jf,L_{b_j}f-\lambda_jf\rangle_b -\langle L_{b_j}f-\lambda_jf,M_{b_j}f-\mu_jf\rangle_b\bigr)\\ &=2i\sum_{j=1}^n\,\operatorname{Im} \langle M_{b_j}f-\mu_jf,L_{b_j}f-\lambda_jf\rangle_b. \end{align*} $$

Hence, on $\mathcal G_b$ ,

$$ \begin{align*}\|f\|_b^2\le\frac1n\sum_{j=1}^n \,\bigl|\langle M_{b_j}f-\mu_jf,L_{b_j}f-\lambda_jf\rangle_b\bigr| \le\frac1n\sum_{j=1}^n\|M_{b_j}f-\mu_jf\|_b\|L_{b_j}f-\lambda_jf\|_b.\end{align*} $$

For equality, we only work out the case indicated in the statement of the theorem, 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 $\langle M_{b_j}f,L_{b_j}f\rangle _b$ is pure imaginary with positive imaginary part. Equality holds in the second inequality if and only if $L_{b_j}f=\beta _jM_{b_j}f$ for some $\beta _j\in \mathbb C$ for each j. The two conditions together imply that equality holds in the uncertainty inequality for an $f^b(z)=\sum \limits _{|\alpha |=0}^\infty f_\alpha ^bz^{\alpha }\in \mathcal G_b$ if and only if $L_{b_j}f^b=i\gamma _jM_{b_j}f^b$ for some $\gamma _j<0$ . Equivalently,

(29) $$ \begin{align} &\frac1{\sqrt{b_1}}\,f_{\epsilon_j}^b +\sum_{|\alpha|=1}^\infty\Biggl(\sqrt{|\alpha|} \,\sqrt{\frac{b_{|\alpha|}}{b_{|\alpha|-1}}}\,f_{\alpha-\epsilon_j}^b +\frac{\alpha_j+1}{\sqrt{|\alpha+1}} \,\sqrt{\frac{b_{|\alpha|}}{b_{|\alpha|+1}}} \,f_{\alpha+\epsilon_j}^b\Biggr)\,z^{\alpha} \\ =&\frac{\gamma_j}{\sqrt{b_1}}\,f_{\epsilon_j}^b +\gamma_j\sum_{|\alpha|=1}^\infty\Biggl(\frac{\alpha_j+1}{\sqrt{|\alpha+1}} \,\sqrt{\frac{b_{|\alpha|}}{b_{|\alpha|+1}}}\,f_{\alpha+\epsilon_j}^b -\sqrt{|\alpha|}\,\sqrt{\frac{b_{|\alpha|}}{b_{|\alpha|-1}}} \,f_{\alpha-\epsilon_j}^b\Biggr)\,z^{\alpha} \nonumber \end{align} $$

for each j.

For each j, the constant terms of (29) give $(1-\gamma _j)f_{\epsilon _j}^b=0$ , which yields

(30) $$ \begin{align} f_{\epsilon_j}^b=0 \end{align} $$

since $\gamma _j<0$ . Setting the coefficients of $z^{\alpha }$ on both sides of (29) equal to each other and using $\gamma _j<0$ , we obtain

(31) $$ \begin{align} f_{\alpha+\epsilon_j}^b=\frac{\gamma_j+1}{\gamma_j-1} \,\frac{\sqrt{|\alpha|}\,\sqrt{|\alpha|+1}}{\alpha_j+1} \,\sqrt{\frac{b_{|\alpha|+1}}{b_{|\alpha|-1}}}\;f_{\alpha-\epsilon_j}^b. \end{align} $$

From now on, we consider only the case $n=2$ . So $\alpha =(\alpha _1,\alpha _2)$ , $j=1,2$ , and we have only $\epsilon _1=(1,0)$ and $\epsilon _2=(0,1)$ . By (30) and (31), $f_{(\text {odd},\text {even})}^b=0$ and $f_{(\text {even},\text {odd})}^b=0$ . Further, by (31), every $f_{(\text {even},\text {even})}^b$ depends on $f_{(0,0)}^b$ . Similar to the proof of Theorem 4.3, with $\alpha =(2l,2m)$ , we obtain

$$ \begin{align*}f_{(2l,2m)}^b=\biggl(\frac{\gamma_1+1}{\gamma_1-1}\biggr)^l \biggl(\frac{\gamma_2+1}{\gamma_2-1}\biggr)^m \frac{\sqrt{(2l+2m)!}}{2^{l+m}\,l!\,m!}\,\sqrt{b_{2l+2m}}\;f_{(0,0)}^b\end{align*} $$

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

(32) $$ \begin{align} f^b(z_1,z_2)=\sum_{l,m=0}^\infty\biggl(\frac{\gamma_1+1}{\gamma_1-1}\biggr)^l \biggl(\frac{\gamma_2+1}{\gamma_2-1}\biggr)^mp_{lm}\,\sqrt{b_{2l+2m}} \;z_1^{2l}z_2^{2m}. \end{align} $$

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

$$ \begin{align*}f_{(2l+1,2m+1)}^b=\biggl(\frac{\gamma_1\!+\!1}{\gamma_1\!-\!1}\biggr)^{\!l} \biggl(\frac{\gamma_2\!+\!1}{\gamma_2\!-\!1}\biggr)^{\!m} \!\frac{\sqrt{(2l+2m+2)!}} {1\!\cdot\!3\cdots(2l\!+\!1)\!\cdot\!1\!\cdot\!3\cdots(2m\!+\!1)} \,\sqrt{\frac{b_{2l+2m+2}}{2b_2}}\;f_{(1,1)}^b\end{align*} $$

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

(33) $$ \begin{align} g^b(z_1,z_2)=\sum_{l,m=0}^\infty\biggl(\frac{\gamma_1+1}{\gamma_1-1}\biggr)^l \biggl(\frac{\gamma_2+1}{\gamma_2-1}\biggr)^mq_{lm} \,\sqrt{\frac{b_{2l+2m+2}}{2b_2}}\;z_1^{2l+1}z_2^{2m+1}. \end{align} $$

We must also check if $f^b$ and $g^b$ belong to $\mathcal G_b$ . It is a routine calculation that

$$ \begin{align*}\frac{\alpha!}{|\alpha|!}\,p_{lm}^2\le1\qquad\text{and}\qquad \frac{\alpha!}{|\alpha|!}\,q_{lm}^2\le1.\end{align*} $$

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

$$ \begin{align*}\le8\sum_{l,m=1}^\infty\biggl(\frac{\gamma_1+1}{\gamma_1-1}\biggr)^{2l} \biggl(\frac{\gamma_2+1}{\gamma_2-1}\biggr)^{2m}(l^2+m^2),\end{align*} $$

which is finite if and only if $\gamma _1<0$ and $\gamma _2<0$ . This shows $f^b\in \mathcal G_b$ for all $\gamma _1<0$ and $\gamma _2<0$ . Similarly, $g^b\in \mathcal G_b$ for all $\gamma _1<0$ and $\gamma _2<0$ .

Specializing to $\mathcal F_q$ , by Definition 3.3,

$$ \begin{align*}L_{q_j}f(z)=\frac{f_{\epsilon_j}}{\sqrt{1+n+q}}+\sum_{|\alpha|=1}^\infty \Biggl(\sqrt{n+q+|\alpha|}\,f_{\alpha-\epsilon_j} +\frac{(1+\alpha_j)f_{\alpha+\epsilon_j}}{\sqrt{1+n+q+|\alpha|}} \Biggr)z^{\alpha}\end{align*} $$

if $q>-(1+n)$ and it equals

$$ \begin{align*}\sqrt{1-n-q}\,f_{\epsilon_j}+\sum_{|\alpha|=1}^\infty \Biggl(\frac{|\alpha|\,f_{\alpha-\epsilon_j}}{\sqrt{-n-q+|\alpha|}} +\sqrt{1-n-q+|\alpha|}\,\frac{1+\alpha_j}{\sqrt{1+|\alpha|}} \,f_{\alpha+\epsilon_j}\Biggr)z^{\alpha}\end{align*} $$

if $q\le -(1+n)$ . 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 $n=1$ : $\mathcal E_q=\mathcal F_{q-1}$ and $\mathcal G_q=\mathcal F_{q-2}$ .

For $n>1$ , there are four important 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 $\mathcal A$ and $q=-(1+n)$ gives the Dirichlet space $\mathcal D$ . The formulas are simplest for $\mathcal A$ since the Drury–Arveson space is special for multivariable operator theory due to $b_k(-n)=1$ for all $k=0,1,2,\ldots $ . So

$$ \begin{align*}L_{(-n)_j}f(z)=f_{\epsilon_j}+\sum_{|\alpha|=1}^\infty \biggl(\sqrt{|\alpha|}\,f_{\alpha-\epsilon_j} +\frac{1+\alpha_j}{\sqrt{1+|\alpha|}}\,f_{\alpha+\epsilon_j}\biggr)z^{\alpha}, \end{align*} $$

and with $q=-n=-2$ ,

$$ \begin{align*}f^{(-2)}(z_1,z_2)=\sum_{l,m=0}^\infty \biggl(\frac{\gamma_1+1}{\gamma_1-1}\biggr)^l \biggl(\frac{\gamma_2+1}{\gamma_2-1}\biggr)^mp_{lm}\;z_1^{2l}z_2^{2m},\end{align*} $$

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