Hardy-Type Space Associated with an Infinite-Dimensional Unitary Matrix Group

Abstract

We investigate an orthogonal system of the homogenous Hilbert-Schmidt polynomials with respect to a probability measure which is invariant under the right action of an infinite-dimensional unitary matrix group. With the help of this system, a corresponding Hardy-type space of square-integrable complex functions is described. An antilinear isomorphism between the Hardy-type space and an associated symmetric Fock space is established.

1. Introduction

We investigate an orthogonal system of the Hilbert-Schmidt polynomials in the space 𝐿²_𝜒 of square-integrable complex functions on the projective limit 𝔘=lim_←𝑈(𝑚) of unitary (𝑚×𝑚)-dimensional matrix groups 𝑈(𝑚)(𝑚∈ℕ), called the space of virtual unitary matrices and endowed with the projective limit measure 𝜒=lim_←𝜒_𝑚 of the probability Haar measures 𝜒_𝑚 on 𝑈(𝑚). The measure 𝜒 on the space 𝔘 is invariant under the right action of the infinite-dimensional unitary group𝑈(∞)×𝑈(∞), where𝑈(∞)=⋃_𝑚𝑈(𝑚).

The space of virtual unitary matrices𝔘was studied by Neretin [1] and Olshanski [2]. This notion relates to D. Pickrell’s space of virtual Grassmannian [3] and to Kerov, Olshanski, and Vershik’s space of virtual permutations [4]. Various spaces of integrable functions with respect to measures that are invariant under infinite-dimensional groups have been widely applied in stochastic processes [5], infinite-dimensional probability [6, 7], complex analysis [8], and so forth.

The main results of the present paper are Theorems 6–7 that describe a Hardy-type subspaceℋ²_𝜒⊂𝐿²_𝜒spanned by the finite type homogenous Hilbert-Schmidt polynomials that are generated by an associated symmetric Fock space.

2. Preliminaries

We consider the following infinite-dimensional unitary matrix groups: 𝑈(∞)={𝑈(𝑚)},𝑈∶𝑚∈ℕ₂(∞)∶=𝑈(∞)𝑈(∞), (1) where𝑈(𝑚)is the group of unitary(𝑚×𝑚)-matrices which is identified with the subgroup in𝑈(𝑚+1)fixing the(𝑚+1)th basis vector. In other words,𝑈(∞)is the group of infinite unitary matrices 𝑢=[𝑢_𝑖𝑗]_{𝑖,𝑗∈ℕ}with finitely many matrix entries𝑢_𝑖𝑗distinct from𝛿_𝑖𝑗. We equip every group 𝑈(𝑚) with the probability Haar measure𝜒_𝑚.

Following [1, 2], every matrix 𝑢_𝑚∈𝑈(𝑚) with𝑚>1, we write in the following block matrix form: 𝑢_𝑚=𝑧_𝑚−1𝑎𝑏𝑡, (2) corresponding to the partition 𝑚=(𝑚−1)+1 so that 𝑧_𝑚−1∈𝑈(𝑚−1) and 𝑡∈ℂ. Over the group 𝑈(∞) (resp., 𝑈(𝑚)) the right action is well defined: 𝑢⋅𝑔=𝑤⁻¹𝑢𝑣, (3) where 𝑢 belongs to 𝑈(∞) (resp., to 𝑈(𝑚)) and 𝑔=(𝑣,𝑤) belongs to 𝑈²(∞) (resp., to 𝑈²(𝑚)∶=𝑈(𝑚)×𝑈(𝑚)). In [1, Proposition 0.1], [2, Lemma 3.1], it was proven that the following Livšic-type mapping: 𝜋^𝑚_𝑚−1∶𝑈(𝑚)∋𝑢_𝑚⟶𝑢_𝑚−1∈𝑈(𝑚−1), (4) such that 𝑧_𝑚−1𝑎⟼𝑧𝑏𝑡^𝑚−1−𝑎(1+𝑡)⁻¹𝑧𝑏∶𝑡≠−1,_𝑚−1∶𝑡=−1, (5) (which is not a group homomorphism) is Borel and surjective onto𝑈(𝑚−1) and commutes with the right action of𝑈²(𝑚−1).

As is known [1, Theorem 1.6], the pullback of the probability Haar measure𝜒_𝑚−1on𝑈(𝑚−1)under the mapping𝜋^𝑚_𝑚−1is the probability Haar measure𝜒_𝑚on𝑈(𝑚), that is, 𝜒_𝑚−1∘𝜋^𝑚_𝑚−1=𝜒_𝑚. (6)

Let 𝑈^′(𝑚)⊂𝑈(𝑚) be the subset of unitary matrices which do not have{−1}, as an eigenvalue. Then, 𝑈^′(𝑚) is open in𝑈(𝑚), and the complement 𝑈(𝑚)⧵𝑈^′(𝑚)is a𝜒_𝑚-negligible set. Moreover (see [2, Lemma 3.11]), the mapping 𝜋^𝑚_𝑚−1∶𝑈^′(𝑚)⟶𝑈^′(𝑚−1) (7) is continuous and surjective.

Consider the projective limits, taken with respect to the surjective Borel projections𝜋^𝑚_𝑚−1and their continuous restrictions𝜋^𝑚_𝑚−1|_{𝑈′(𝑚)}, respectively, 𝔘=lim_⟵𝑈(𝑚),𝔘^′=lim_⟵𝑈^′(𝑚), (8) called the spaces of virtual unitary matrices. Notice that𝔘is a Borel subset in the Cartesian product ⨉_𝑚∈ℕ𝑈(𝑚)={𝑢=(𝑢_𝑚)∶𝑢_𝑚∈𝑈(𝑚)}endowed with the product topology, because all mapping𝜋^𝑚_𝑚−1are Borel. Moreover, the canonical projections 𝜋_𝑚∶𝔘⟶𝑈(𝑚),𝜋_𝑚∶𝔘^′⟶𝑈^′(𝑚), (9) such that𝜋_𝑚−1=𝜋^𝑚_𝑚−1∘𝜋_𝑚, are surjective by surjectivity of𝜋^𝑚_𝑚−1and𝜋^𝑚_𝑚−1|_{𝑈′(𝑚)}.

Following [2, Lemma 4.8], [1, Section 3.1], with the help of the Kolmogorov consistent theorem, we uniquely define a probability measure𝜒on𝔘^′as the projective limit under the mapping (6), 𝜒=lim_⟵𝜒_𝑚, (10) which satisfies the equality𝜒=𝜒_𝑚∘𝜋_𝑚for all𝑚∈ℕ. On𝔘⧵𝔘^′, the measure𝜒is zero, because𝜒_𝑚is zero on𝑈(𝑚)⧵𝑈^′(𝑚)for all𝑚∈ℕ.

Using (3), right action of the group𝑈²(∞)on the space of virtual unitary matrices𝔘can be defined (see [2, Definition 4.5]) as follows: 𝜋_𝑚𝑢⋅𝑔=𝑤⁻¹𝜋_𝑚(𝑢)𝑣,𝑢∈𝔘, (11) where 𝑚 is so large that 𝑔=(𝑣,𝑤)∈𝑈²(𝑚).

The canonical dense embedding 𝚤∶𝑈(∞)↬𝔘 to any element𝑢_𝑚∈𝑈(𝑚)assigns the unique sequence 𝑢=(𝑢_𝑙)_𝑙∈ℕ, such that 𝚤∶𝑈(𝑚)∋𝑢_𝑚⟼𝑢_𝑙𝑢∈𝔘,_𝑙=𝜋_𝑙^𝑙+1∘⋯∘𝜋^𝑚_𝑚−1𝑢_𝑚𝑢∶𝑙<𝑚,_𝑚𝑢∶𝑙=𝑚,_𝑚00𝟙_𝑙−𝑚∶𝑙>𝑚, (12) where𝟙_𝑙−𝑚is the unit in𝑈(𝑙−𝑚). So, the image𝚤∘𝑈(∞)consists of stabilizing sequences in𝔘(see [2, Section 4]).

3. Invariant Probability Measure

In what follows, we will endow the space of virtual unitary matrices𝔘with the measure𝜒=lim_←𝜒_𝑚. A complex function on𝔘is called cylindrical [2, Definition 4.5] if it has the following form: 𝑓(𝑢)=𝑓_𝑚∘𝜋_𝑚(𝑢),𝑢∈𝔘, (13) for a certain𝑚∈ℕand a certain complex function𝑓_𝑚on𝑈(𝑚).

Any continuous bounded function𝑓on𝔘^′has a unique𝜒-essentially bounded extension on𝔘, because the set𝔘⧵𝔘^′is𝜒-negligible. Therefore, if the function𝑈^′(𝑚)∋𝜋_𝑚(𝑢)↦𝑓_𝑚[𝜋_𝑚(𝑢)]in the definition (13) is continuous and bounded, then the corresponding cylindrical function𝑓is𝜒 essentially bounded.

Byℒ^∞_𝜒, we denote closure of the algebraic hull of all cylindrical𝜒-essentially bounded functions (13) with respect to the following norm: 𝑓_ℒ∞𝜒=esssup_𝑢∈𝔘𝑓(𝑢). (14)

Lemma 1. The measure𝜒=lim_←𝜒_𝑚on𝔘is a Radon probability measure such that _𝔘𝑓𝑢⋅𝑔𝑑𝜒(𝑢)=_𝔘𝑓(𝑢)𝑑𝜒(𝑢), (15) for all𝑔∈𝑈²(∞)and𝑓∈ℒ^∞_𝜒. For any compact set𝐾⊂𝑈(𝑚)the following equality holds: 𝜒∘𝚤(𝐾)=𝜒_𝑚(𝐾). (16)

Proof. Recall the Prohorov criterion, which is adapted to our notation (see [9, Chapter IX.4.2, Theorem 1] or [6, Theorem 6]): there exists a Radon probability measure𝜒^′on𝔘^′such that 𝜒^′=𝜒_𝑚∘𝜋_𝑚𝔘′∀𝑚∈ℕ, (17) if and only if for every𝜀>0there exists a compact set𝒦in𝔘^′such that the following inequality 𝜒_𝑚∘𝜋_𝑚(𝒦)≥1−𝜀∀𝑚∈ℕ (18) holds; in this case, 𝜒^′ is uniquely determined by means of the formula𝜒^′(𝒦)=inf_𝑚∈ℕ(𝜒_𝑚∘𝜋_𝑚)(𝒦), where𝒦is a compact set in𝔘^′.
Let𝐾_𝑛⊂𝑈^′(𝑛)be a compact set with a fixed 𝑛. Putting𝐾_𝑛−1=𝜋^𝑛_𝑛−1(𝐾_𝑛), we have 𝜒_𝑛−1𝐾_𝑛−1=𝜒_𝑛−1∘𝜋^𝑛_𝑛−1𝐾_𝑛=𝜒_𝑛𝐾_𝑛. (19) On the other hand, if we put𝐾_𝑛+1=^𝐾𝑛0₀₁, then via (6), 𝜒_𝑛+1𝐾_𝑛+1=𝜒_𝑛∘𝜋_𝑛^𝑛+1𝐾_𝑛+1=𝜒_𝑛∘𝜋_𝑛^𝑛+1𝐾_𝑛001=𝜒_𝑛𝐾_𝑛. (20) As a consequence, the compact set𝒦=(𝐾_𝑚)in𝔘^′, generated by a compact set𝐾_𝑛⊂𝑈^′(𝑛)with the help of mappings𝜋^𝑛_𝑛−1, satisfies the following condition: 𝜒_𝑛𝐾_𝑛=𝜒_𝑚𝐾_𝑚∀𝑚∈ℕ. (21)
The probability Haar measure𝜒_𝑛is regular on𝑈(𝑛), and the complement𝑈(𝑛)⧵𝑈^′(𝑛)is a negligible set. Hence, if𝐾_𝑛runs over all compact sets in𝑈^′(𝑛), then sup_{𝐾𝑛⊂𝑈′(𝑛)}𝜒_𝑛𝐾_𝑛=1. (22) Therefore, for every𝜀>0there exists a compact set𝐾_𝑛⊂𝑈^′(𝑛)such that𝜒_𝑛(𝐾_𝑛)≥1−𝜀. From (21), it follows that for every𝜀>0the compact set𝒦satisfies the hypothesis of Prohorov’s criterion: 𝜒_𝑚∘𝜋_𝑚(𝒦)=𝜒_𝑚𝐾_𝑚≥1−𝜀∀𝑚∈ℕ. (23) So, in view of this criterion, there exists a unique Radon probability measure𝜒^′on𝔘^′which satisfies the condition (17). However, on the projective limits𝔘^′=lim_←𝑈^′(𝑚), there exists a unique𝑈²(∞)-invariant Radon measure𝜒, determined by the equality (15). Using the uniqueness property of projective limits, we obtain𝜒^′=𝜒. The measure𝜒on𝔘⧵𝔘^′is defined to be zero, because𝜒_𝑚is zero on𝑈(𝑚)⧵𝑈^′(𝑚).
As a consequence of (21), we obtain (16), because 𝜒(𝒦)=inf_𝑚∈ℕ𝜒_𝑚𝐾_𝑚=𝜒_𝑛𝐾_𝑛. (24)
As is known [1, Proposition 3.2], the measure𝜒is𝑈²(∞)-invariant under the right actions (11) on the space𝔘. Hence, for every𝑓∈ℒ^∞_𝜒, the equality (15) holds.

4. Shift Groups

Consider that in the spaceℒ^∞_𝜒, the group of shifts 𝑄_𝑔𝑓(𝑢)=𝑓𝑢⋅𝑔,𝑔∈𝑈²(∞)𝑢∈𝔘, (25) is generated by the right action of𝑈²(∞)over𝔘. Choosing instead of𝑈(∞)a compact subgroup 𝑈(𝑚)or the compact subgroups 𝑈₀=𝑔₀(𝜗)()(,𝑈=exp𝔦𝜗∶𝜗∈−𝜋,𝜋]_𝑗(𝑚)=𝑔_𝑚𝑗(𝜗)=𝟙_𝑗−1⊗exp(𝔦𝜗)⊗𝟙_𝑚−𝑗∶𝜗∈(−𝜋,𝜋]𝑗=1,…,𝑚, (26) we obtain the corresponding subgroups of shifts𝑄_𝑔with elements𝑔∈𝑈²(𝑚)or with elements 𝑔₀(𝜗)∈𝑈²₀and𝑔_𝑚𝑗(𝜗)∈𝑈²_𝑗(𝑚), respectively. Here, 𝟙_𝑚means the unit element in𝑈(𝑚).

Lemma 2. For any𝑓∈ℒ^∞_𝜒the following equalities: _𝔘𝑓𝑑𝜒=_𝔘𝑑𝜒(𝑢)_𝑈2(𝑚)𝑄_𝑔𝑓(𝑢)𝑑𝜒_𝑚⊗𝜒_𝑚𝑔,_𝔘1𝑓𝑑𝜒=2𝜋_𝔘𝑑𝜒(𝑢)^𝜋_−𝜋𝑄_𝑔(𝜗)𝑓(𝑢)𝑑𝜗, (27)(28) with𝑔(𝜗)∈𝑈²₀or𝑈²_𝑗(𝑚)hold.

Proof. For any𝑓∈ℒ^∞_𝜒, the function (𝑢,𝑔)↦𝑄_𝑔𝑓(𝑢)=𝑓(𝑢⋅𝑔) is integrable on the Cartesian product 𝔘×𝑈²(𝑚). By the Fubini theorem, we obtain 𝑑𝜒(_𝔘𝑢)_𝑈2(𝑚)𝑄_𝑔𝑓(𝑢)𝑑𝜒_𝑚⊗𝜒_𝑚𝑔=_𝑈2(𝑚)𝑑𝜒_𝑚⊗𝜒_𝑚𝑔_𝔘𝑄_𝑔𝑓(𝑢)𝑑𝜒(𝑢). (29)
This equality yields the required formula (27), because the internal integral on the right-hand side is independent of𝑔and∫_𝑈2(𝑚)𝑑(𝜒_𝑚⊗𝜒_𝑚)=1. In turn, putting instead of𝑈(𝑚)the subgroups𝑈₀and 𝑈_𝑗(𝑚), we obtain equalities (28).

5. The Homogeneous Hilbert-Schmidt Polynomials

Consider the countable orthogonal Hilbertian sum 𝖤∶=_𝑚∈ℕℂ^𝑚=𝑥=𝑥_𝑚∶𝑥_𝑚∈ℂ^𝑚,‖𝑥‖_𝖤<∞, (30) with the scalar product⟨𝑥∣𝑦⟩_𝖤=∑_𝑚⟨𝑥_𝑚∣𝑦_𝑚⟩_ℂ𝑚, where every coordinate𝑥_𝑚∈ℂ^𝑚is identified with its image (0,…,0,𝑥_𝑚,0,…)∈𝖤 under the embeddingℂ^𝑚↬𝖤.

Let ⊗^𝑛_𝔥𝖤 stand for the complete 𝑛th tensor power of the Hilbert subspace 𝖤, endowed with the Hilbertian scalar product and norm, respectively, ⟨𝑥₁⊗⋯⊗𝑥_𝑛∣𝜓_𝑛⟩_{⊗𝑛𝔥𝖤}=_𝑗⟨𝑥₁∣𝑦_1𝑗⟩_𝖤…⟨𝑥_𝑛∣𝑦_𝑛𝑗⟩_𝖤,𝜓_{𝑛⊗𝑛𝔥𝖤}=𝜓_𝑛∣𝜓_𝑛⊗^1/2_𝑛𝔥𝖤, (31) where 𝑥₁⊗⋯⊗𝑥_𝑛, 𝑦_1𝑗⊗⋯⊗𝑦_𝑛𝑗∈⊗^𝑛_𝔥𝖤 with𝑥_𝑡𝑗,𝑦_𝑡𝑗∈𝖤for all 𝑡=1,…,𝑛 and 𝜓_𝑛=∑_𝑗𝑦_1𝑗⊗⋯⊗𝑦_𝑛𝑗 denotes a finite sum. Put ⊗⁰_𝔥𝖤=ℂ. We use the following short denotation: 𝑥^⊗𝑛=𝑥⊗⋯⊗𝑥,𝑥∈𝖤. (32)

Replacing the space𝖤by the subspaceℂ^𝑚, we similarly define the tensor product⊗^𝑛_𝔥ℂ^𝑚. There is the unitary embedding ⊗^𝑛_𝔥ℂ^𝑚↬⊗^𝑛_𝔥𝖤. If 𝑚=1, then ⊗^𝑛_𝔥ℂ=ℂ.

For any finite sum𝜓_𝑛=∑_𝑗𝑦_1𝑗⊗⋯⊗𝑦_𝑛𝑗from the space⊗^𝑛_𝔥ℂ^𝑚(or⊗^𝑛_𝔥𝖤), we can to define the finite type 𝑛-homogeneous Hilbert-Schmidt polynomials: ℂ^𝑚∋𝑥⟼𝑥^⊗𝑛∣𝜓_{𝑛⊗𝑛𝔥ℂ𝑚}=_𝑗^𝑛_𝑡=1𝑥∣𝑦_{𝑡𝑗ℂ𝑚}. (33)

Consider the canonical orthonormal bases: ℰℂ^𝑚=𝔢_𝑚1,…,𝔢_𝑚𝑚inℂ^𝑚,ℰ(𝖤)=ℰℂ^𝑚∶𝑚∈ℕin𝖤, (34) where𝔢_𝑚𝑙=(0,…,0,1^𝑙,0,…,0)_𝑚.

If𝔰∶{1,…,𝑛}↦{𝔰(1),…,𝔰(𝑛)}runs over all𝑛-elements permutations𝔖(𝑛), then the symmetric𝑛th tensor power⊙^𝑛_𝔥ℂ^𝑚is defined to be a codomain of the symmetrization mapping: ⊗^𝑛_𝔥ℂ^𝑚∋𝑥₁⊗⋯⊗𝑥_𝑛⟼𝑥₁⊙⋯⊙𝑥_𝑛,𝑥₁⊙⋯⊙𝑥_𝑛1∶=𝑛!_{𝔰∈𝔖(𝑛)}𝑥_𝔰(1)⊗⋯⊗𝑥_𝔰(𝑛), (35) which is an orthogonal projector. Similarly, the symmetric𝑛th tensor power⊙^𝑛_𝔥𝖤can be defined. Clearly, ⊙^𝑛_𝔥ℂ^𝑚is a closed subspace in⊙^𝑛_𝔥𝖤.

Given a pair of numbers(𝑚,𝑛)∈ℕ×ℤ₊, we consider the𝑛-fold tensor power of the canonical mapping 𝜋_𝑚∶𝔘∋𝑢↦𝜋_𝑚(𝑢)∈𝑈(𝑚), 𝔘∋𝑢⟼𝜋_𝑚^⊗𝑛(𝑢)∈ℒ⊙^𝑛_𝔥ℂ^𝑚, (36) where 𝜋_𝑚^⊗𝑛(𝑢)∶=𝜋_𝑚(𝑢)⊗⋯⊗𝜋_𝑚(𝑢)_𝑛. If𝑛=0, we put𝜋_𝑚^⊗0(𝑢)=1for all𝑢∈𝔘and𝑚∈ℕ. The mapping (36) is Borel and has a continuous restriction to𝔘^′, because𝜋_𝑚has the same property (see Section 2).

Let 𝔞_𝑚∈ℂ^𝑚be an arbitrary fixed element such that ‖𝔞_𝑚‖_ℂ^𝑚=1. Then, 𝔞_𝑚^⊗𝑛∈⊙^𝑛_𝔥ℂ^𝑚. Using the mapping (36), we can write 𝜋_𝑚^⊗𝑛(𝑢)𝔞_𝑚^⊗𝑛=𝜋_𝑚(𝑢)𝔞_𝑚⊗⋯⊗𝜋_𝑚(𝑢)_𝑚𝑛𝔞. (37) To any 𝑛-homogeneous Hilbert-Schmidt polynomial (33), there corresponds the function 𝜓^∗_𝑛(𝑢)∶=𝜋_𝑚^⊗𝑛(𝑢)𝔞_𝑚^⊗𝑛∣𝜓_{𝑛⊗𝑛𝔥ℂ𝑚}=_𝑗^𝑛_𝑡=1𝜋_𝑚(𝑢)𝔞_𝑚∣𝑦_{𝑡𝑗ℂ𝑚} (38) of the variable𝑢∈𝔘. Any cylindrical function of the form𝔘∋𝑢↦⟨[𝜋_𝑚(𝑢)](𝔞_𝑚)∣𝑦_𝑡𝑗⟩_ℂ𝑚has a continuous bounded restriction to𝔘^′. Therefore, it is𝜒-essentially bounded on𝔘, because𝔘⧵𝔘^′ is a𝜒-negligible set. Consequently,𝜓^∗_𝑛∈𝐿^∞_𝜒and𝜓^∗_𝑛|_𝔘′ is continuous and bounded.

Definition 3. We define𝒫^𝑛_𝔥(ℂ^𝑚)to be the space of all functions𝜓^∗_𝑛of the variable𝑢∈𝔘, determined by the finite type𝑛-homogeneous Hilbert-Schmidt polynomials (33).

Lemma 4. For any element 𝔞_𝑚∈ℂ^𝑚 such that‖𝔞_𝑚‖_ℂ^𝑚=1 the set 𝖲^𝑚=𝑥=𝜋_𝑚(𝑢)𝔞_𝑚∶𝑢∈𝔘 (39) coincides with the unit sphere inℂ^𝑚. As a consequence, the one-to-one antilinear corresponding ⊙^𝑛_𝔥ℂ^𝑚∋𝜓_𝑛⇄𝜓^∗_𝑛∈𝒫^𝑛_𝔥ℂ^𝑚. (40) Holds, and any function 𝜓^∗_𝑛 is independent of the choice of an element 𝔞_𝑚∈𝖲^𝑚.

Proof. Suppose, on the contrary, that there is an element𝜓_𝑛∈⊙^𝑛_𝔥ℂ^𝑚such that⟨𝑥^⊗𝑛∣𝜓_𝑛⟩_⊗^𝑛_𝔥ℂ^𝑚=0for all 𝑥=[𝜋_𝑚(𝑢)](𝔞_𝑚)∈𝖲^𝑚with𝑢∈𝔘. The mapping 𝜋_𝑚∶𝔘∋𝑢⟼𝜋_𝑚(𝑢)∈𝑈(𝑚) (41) is surjective by surjectivity of the mapping𝜋_𝑚(see [2, Lemma 3.1]). Hence, the set 𝖲^𝑚 coincides with the unit sphere in ℂ^𝑚and is independent on the choice of an element 𝔞_𝑚. By 𝑛-homogeneity, we have ⟨𝑥^⊗𝑛∣𝜓_𝑛⟩_⊗^𝑛_𝔥ℂ^𝑚=0 for all 𝑥∈ℂ^𝑚.
Apply the following polarization formula for symmetric tensor products (see, e.g., [10, Section 1.5]): 𝑧₁⊙⋯⊙𝑧_𝑛=12^𝑛𝑛!_{1≤𝑡≤𝑛𝛿𝑡=±1}𝛿₁⋯𝛿_𝑛𝑥^⊗𝑛, (42) with𝑥=∑^𝑛_𝑡=1𝛿_𝑡𝑧_𝑡∈ℂ^𝑚, which is valid for all𝑧₁,…,𝑧_𝑛∈ℂ^𝑚. It follows that ⟨𝑧₁⊙⋯⊙𝑧_𝑛∣𝜓_𝑛⟩_⊗^𝑛_𝔥ℂ^𝑚=0for all elements𝑧₁,…,𝑧_𝑛∈ℂ^𝑚. Hence, 𝜓_𝑛=0, because the subset of all elements𝑧₁⊙⋯⊙𝑧_𝑛is total in⊙^𝑛_𝔥ℂ^𝑚. As a consequence, the subset 𝑥^⊗𝑛=𝜋_𝑚^⊗𝑛(𝑢)𝔞_𝑚^⊗𝑛∶𝑢∈𝔘 (43) is also total in⊙^𝑛_𝔥ℂ^𝑚. It immediately yields the correspondence (40).

Consider the symmetric Fock space 𝖥 and its closed subspace 𝖥_𝑚, where 𝖥∶=ℂ⊕𝖤⊕⊙²_𝔥𝖤⊕⊙³_𝔥𝖤𝖥⊕⋯,_𝑚∶=ℂ⊕ℂ^𝑚⊕⊙²_𝔥ℂ^𝑚⊕⊙³_𝔥ℂ^𝑚⊕⋯. (44) We will use the following notations: (𝑚)(),𝑘∶=𝑚1,…,𝑚𝑚_(𝑚)∶=𝑘_𝑚1,…,𝑘_𝑚𝑚∈ℤ^𝑚₊,𝑘_(𝑚)∶=𝑘_𝑚1+⋯+𝑘_𝑚𝑚,𝑘_(𝑚)!∶=𝑘_𝑚1!⋅…⋅𝑘_𝑚𝑚!. (45) As is well known (see, e.g., [11]), the system of symmetric tensor elements, indexed by the set 𝑘_(𝑚), ℰ⊙^𝑛_𝔥ℂ^𝑚=𝔢^{⊗𝑘(𝑚)}_(𝑚)=𝔢^⊗𝑘𝑚1_𝑚1⊙⋯⊙𝔢^{⊗𝑘𝑚𝑚}_𝑚𝑚∶𝑘_(𝑚)∈ℤ^𝑚₊;𝑘_(𝑚)=𝑛 (46) forms an orthogonal basis in the subspace ⊙^𝑛_𝔥ℂ^𝑚⊂𝖥_𝑚. (47) We will also use the following notations: [𝑚]),()(,{𝑘}𝑘∶={(1121,22,…,𝑚1,…,𝑚𝑚)}∶=₍₁₎,…,𝑘_(𝑚)∈^𝑚_𝑟=1ℤ^𝑟₊,|{𝑘}|∶=𝑘₍₁₎+⋯+𝑘_(𝑚),{𝑘}!∶=𝑘₍₁₎!⋅…⋅𝑘_(𝑚)!. (48) Then, the system of symmetric tensor elements with a fixed𝑛, indexed by the sets[𝑚]and{𝑘}, ℰ_𝑛=_𝑚∈ℕ𝔢_[𝑚]^⊗{𝑘}=𝔢^⊗𝑘(1)₍₁₎⊙⋯⊙𝔢^{⊗𝑘(𝑚)}_(𝑚)∶𝔢^⊗𝑘(1)₍₁₎∈ℰ⊙^|𝑘(1)|_𝔥ℂ,…,𝔢^{⊗𝑘(𝑚)}_(𝑚)∈ℰ⊙^|𝑘(𝑚)|_𝔥ℂ^𝑚withfixed|{𝑘}|=𝑛, (49) forms an orthogonal basis in the subspace⊙^𝑛_𝔥𝖤⊂𝖥. Thus, the system ℰ=ℰ_𝑛∶𝑛∈ℤ₊ (50) forms an orthogonal basis in the symmetric Fock space 𝖥.

By virtue of the one-to-one mapping (40), the system of symmetric tensor elementsℰ(⊙^𝑛_𝔥ℂ^𝑚)uniquely defines the following corresponding system: ℰ^∗_𝑚,𝑛⊂𝒫^𝑛_𝔥ℂ^𝑚, (51) of the following 𝜒_𝑚-integrable cylindrical functions: 𝔢^{∗𝑘(𝑚)}_(𝑚)(𝑢)∶=𝜋_𝑚^⊗𝑛(𝑢)𝔢^⊗𝑛_𝑚1∣𝔢^{⊗𝑘(𝑚)}_{(𝑚)⊗𝑛𝔥ℂ𝑚}=^𝑚_𝑟=1𝜋_𝑚∘𝑢𝔢_𝑚1∣𝔢_𝑚𝑟^𝑘𝑚𝑟_ℂ𝑚, (52) of the variable𝑢∈𝔘, where we take𝔞_𝑚=𝔢_𝑚1. Consider the system of functions of the variable𝑢∈𝔘, ℰ^∗_𝑛=_𝑚∈ℕ𝔢_[𝑚]^∗{𝑘}=𝔢^∗𝑘(1)₍₁₎⋅⋯⋅𝔢^{∗𝑘(𝑚)}_(𝑚)∶𝔢^∗𝑘(1)₍₁₎∈ℰ^∗_𝑘1,(1),…,𝔢^{∗𝑘(𝑚)}_(𝑚)∈ℰ^∗_{𝑘𝑚,(𝑚)}withfixed|{𝑘}|=𝑛, (53) generated by the system of symmetric tensor elementsℰ_𝑛. All these functions belong to the spaceℒ^∞_𝜒by their definition. Denote ℰ^∗=ℰ^∗_𝑛∶𝑛∈ℤ₊,ℰ^∗_𝑚=ℰ^∗_𝑚,𝑛∶𝑛∈ℤ₊. (54)

6. The Hardy-Type Space

Let𝐿²_𝜒be the space of square𝜒-integrable complex functions𝑓on the space of virtual matrices𝔘. Since 𝜒is a probability measure, the embeddingℒ^∞_𝜒⊂𝐿²_𝜒holds and 𝑓_𝐿2𝜒≤esssup_𝑢∈𝔘𝑓(𝑢),𝑓∈ℒ^∞_𝜒. (55)

Denote byℋ²_𝜒𝑚the𝐿²_𝜒-closure of complex linear spans of the subsystemℰ^∗_𝑚. As is well known (see, e.g., [12, Theorem 5.6.8]), the spaceℋ²_𝜒𝑚is isomorphic to the classic Hardy spaceℋ²_𝜒𝑚(𝖡^𝑚)of analytic complex functions on the open unit ball 𝖡^𝑚={𝑥_𝑚∈ℂ^𝑚∶‖𝑥_𝑚‖_ℂ^𝑚<1}. Therefore, the following more general definition seems natural (see, also [8]).

Definition 5. The Hardy-type spaceℋ²_𝜒on the space of virtual unitary matrices𝔘is defined to be the𝐿²_𝜒-closure of the complex linear span of the systemℰ^∗.

Theorem 6. The systemℰ^∗of all functions𝔢^∗{𝑘}_[𝑚]=𝔢^∗𝑘(1)₍₁₎⋅⋯⋅𝔢^{∗𝑘(𝑚)}_(𝑚)with𝑚∈ℕ, such that 𝔢^{∗𝑘(𝑟)}_(𝑟)∈ℰ^∗_{𝑟,|𝑘(𝑟)|}as𝑟=1,…,𝑚, forms an orthogonal basis in the Hardy-type spacesℋ²_𝜒with norms 𝔢_[𝑚]^∗{𝑘}_𝐿2𝜒=^𝑚_𝑟=1(𝑟−1)!(𝑘)_𝑟!𝑟−1+(𝑘)_𝑟!^1/2. (56)

Proof. If|{𝑘}|≠|{𝑞}|, then from (28), it follows that _𝔘𝔢_[𝑚]^∗{𝑘}⋅𝔢_[𝑛]^∗{𝑞}=𝑑𝜒_𝔘𝔢_[𝑚]^∗{𝑘}exp(𝔦𝜗)𝑢⋅𝔢_[𝑛]^∗{𝑞}()𝑢(𝑢)=1exp𝔦𝜗𝑑𝜒2𝜋_𝔘𝔢_[𝑚]^∗{𝑘}𝔢_[𝑛]^∗{𝑞}𝑑𝜒^𝜋_−𝜋exp𝔦|{𝑘}|−𝑞𝜗𝑑𝜗=0. (57) So, 𝔢^∗{𝑘}_[𝑚]⟂𝔢^∗{𝑞}_[𝑛] in the space𝐿²_𝜒if|{𝑘}|≠|{𝑞}|for all indices[𝑚],[𝑛].
Let |{𝑘}|=|{𝑞}| and 𝑚>𝑛 for definiteness. If the elements 𝔢^∗{𝑘}_[𝑚] and 𝔢^∗{𝑞}_[𝑛] are different, then there exists a subindex 𝑚𝑠∈{11,21,22,…,𝑚1,…,𝑚𝑚} in the block-index [𝑚]=[(11),(21,22),…,(𝑚1,…,𝑚𝑚)] such that 𝑚𝑠∉{11,21,22,…,𝑛1,…,𝑛𝑛}, where [𝑛]=[(11),(21,22),…,(𝑛1,…,𝑛𝑛)]. The formula (28) implies that for the group of shifts 𝑄_{𝑔𝑚𝑠(𝜗)} generated by elements 𝑔_𝑚𝑠(𝜗)∈𝑈²_𝑠(𝑚)with the subindex𝑚𝑠, _𝔘𝔢_[𝑚]^∗{𝑘}⋅𝔢_[𝑛]^∗{𝑞}=𝑑𝜒_𝔘𝑄_{𝑔𝑚𝑠(𝜗)}𝔢_[𝑚]^∗{𝑘}⋅𝑄_{𝑔𝑚𝑠(𝜗)}𝔢_[𝑛]^∗{𝑞}=1𝑑𝜒2𝜋_𝔘𝔢_[𝑚]^∗{𝑘}⋅𝔢_[𝑛]^∗{𝑞}𝑑𝜒^𝜋_−𝜋exp𝔦𝑘_𝑚𝑠𝜗𝑑𝜗=0. (58) Hence,𝔢^∗{𝑘}_[𝑚]⟂𝔢^∗{𝑞}_[𝑛]in𝐿²_𝜒.
Let now|{𝑘}|=|{𝑞}|and𝑚=𝑛. If𝔢^∗{𝑘}_[𝑚]≠𝔢^∗{𝑞}_[𝑛], then{𝑘}≠{𝑞}. Hence, there exists a sub-index𝑟𝑠in the block-index [𝑚]=[𝑛] such that𝑘_𝑟𝑠≠𝑞_𝑟𝑠. Similarly as previous mentioned, applying the formula (28) to the group of shifts𝑄_{𝑔𝑟𝑠(𝜗)}generated by elements𝑔_𝑟𝑠(𝜗)∈𝑈²_𝑠(𝑟)with the sub-index𝑟𝑠, we get _𝔘𝔢_[𝑚]^∗{𝑘}⋅𝔢_[𝑛]^∗{𝑞}=1𝑑𝜒2𝜋_𝔘𝔢_[𝑚]^∗{𝑘}𝔢_[𝑛]^∗{𝑞}𝑑𝜒^𝜋_−𝜋exp𝔦𝑘_𝑟𝑠−𝑞_𝑟𝑠𝜗𝑑𝜗=0. (59) Hence, in this case also𝔢^∗{𝑘}_[𝑚]⟂𝔢^∗{𝑞}_[𝑛]under the measure𝜒.
Let𝑔_𝑟=(𝟙_𝑟,𝑤_𝑟)∈𝑈²(𝑟)and𝑢∈𝔘. Using (11) and (52), we have _𝑈2(𝑟)𝑄_𝑔𝑟𝔢^{∗(𝑘)𝑟}_(𝑟)²(𝑢)𝑑𝜒_𝑟⊗𝜒_𝑟𝑔_𝑟=^𝑟_{𝑈(𝑟)𝑙=1}𝑤_𝑟⁻¹𝜋_𝑟(𝑢)𝔢_𝑟1∣𝔢_𝑟𝑙^𝑘𝑟𝑙_ℂ𝑟²𝑑𝜒_𝑟𝑤_𝑟. (60) However, the previous integral with the Haar measure𝜒_𝑟is independent of𝜋_𝑟(𝑢)∈𝑈(𝑟). It follows that _𝑈2(𝑟)𝑄_𝑔𝑟𝔢^{∗(𝑘)𝑟}_(𝑟)²(𝑢)𝑑𝜒_𝑟⊗𝜒_𝑟𝑔_𝑟=^𝑟_{𝑈(𝑟)𝑙=1}𝑤_𝑟⁻¹𝔢_𝑟1∣𝔢_𝑟𝑙^𝑘𝑟𝑙_ℂ𝑟²𝑑𝜒_𝑟𝑤_𝑟=(𝑟−1)!(𝑘)_𝑟!𝑟−1+(𝑘)_𝑟!=𝔢^{∗(𝑘)𝑟}_(𝑟)²_{𝐿2𝜒𝑟} (61) by the well-known formula [12, Section 1.4.9]. Using the formula (27)𝑚-times for𝑟=1,…,𝑚, we get _𝔘𝔢_[𝑚]^∗{𝑘}2=𝑑𝜒_𝔘𝑑𝜒(𝑢)^𝑚_{𝑟=1𝑈2(𝑟)}𝑄_𝑔𝑟𝔢^{∗(𝑘)𝑟}_(𝑟)²(𝑢)𝑑𝜒_𝑟⊗𝜒_𝑟𝑔_𝑟=^𝑚_𝑟=1𝔢^{∗𝑘(𝑟)}_(𝑟)²_{𝐿2𝜒𝑟}, (62) because∫_𝔘𝑑𝜒=1. It follows that 𝔢_[𝑚]^∗{𝑘}2_𝐿2𝜒=^𝑚_𝑟=1𝔢^{∗𝑘(𝑟)}_(𝑟)²_{𝐿2𝜒𝑟}=^𝑚_𝑟=1(𝑟−1)!(𝑘)_𝑟!𝑟−1+(𝑘)_𝑟!, (63) for all𝔢^∗{𝑘}_[𝑚]=𝔢^{∗𝑘(𝑚)}₍₁₎⋅⋯⋅𝔢^{∗𝑘(𝑚)}_(𝑚).

As is known (see, e.g., [11]), the systemℰ_𝑚of symmetric tensors𝔢^{⊗(𝑘)𝑚}_(𝑚)with a fixed𝑚forms an orthogonal basis in the symmetric Fock space𝖥_𝑚with norms‖𝔢^{⊗(𝑘)𝑚}_(𝑚)‖_𝖥𝑚=(𝑘)_𝑚!/|(𝑘)_𝑚|!. Similarly, the systemℰof symmetric tensors𝔢^⊗{𝑘}_[𝑚]=𝔢^⊗(𝑘)1₍₁₎⊙⋯⊙𝔢^{⊗𝑘(𝑚)}_(𝑚)with all𝑚∈ℕ, such that𝔢^{⊗(𝑘)𝑟}_(𝑟)∈ℰ_{𝑟,|(𝑘)𝑟|}as𝑟=1,…,𝑚, forms an orthogonal basis in the symmetric Fock space𝖥with norms ‖𝔢^⊗{𝑘}_[𝑚]‖_𝖥={𝑘}!/|{𝑘}|!.

Combining Lemma 4, Theorem 6, and [12, Theorem 5.6.8], we obtain the following.

Theorem 7. Antilinear extensions of the one-to-one mappings between the orthonormal bases 𝔢^{⊗(𝑘)𝑚}_(𝑚)𝔢^{⊗(𝑘)𝑚}_{(𝑚)𝖥𝑚}⇄𝔢^{∗(𝑘)𝑚}_(𝑚)𝔢^{∗(𝑘)𝑚}_{(𝑚)𝐿2𝜒𝑚},𝔢_[𝑚]^⊗{𝑘}𝔢_[𝑚]^⊗{𝑘}_𝖥⇄𝔢_[𝑚]^∗{𝑘}𝔢_[𝑚]^∗{𝑘}_𝐿2𝜒, (64) uniquely define the corresponding anti-linear isometric isomorphisms 𝖥_𝑚≃ℋ²_𝜒𝑚𝖡^𝑚,𝖥≃ℋ²_𝜒. (65)

Reasoning by analogy with [8, Proposition 6.1 and Theorem 7.1], it is easy to show that the Hardy space ℋ²_𝜒possesses the reproducing kernel of a Cauchy type ℭ(𝑣,𝑢)=_{𝑛∈ℤ+𝑘|{}|=𝑛}𝔢_[𝑚]^∗{𝑘}(𝑣)𝔢_[𝑚]^∗{𝑘}(𝑢)𝔢_[𝑚]^∗{𝑘}2_𝐿2𝜒=^∞_𝑚=11−𝜋_𝑚∘𝑣𝔢_𝑚1|𝜋_𝑚∘𝑢𝔢_𝑚1𝖤^−𝑚, (66) with𝑢,𝑣∈𝔘, where the sum∑_{|{𝑘}|=𝑛}is over all indices{𝑘}∈{⨉^𝑚_𝑟=1ℤ^𝑟₊∶𝑚∈ℕ}such that |{𝑘}|=𝑛. As a consequence, the integral representation of any function𝑓∈ℋ²_𝜒, 𝑓(𝜆𝑣)=_𝔘𝑓(𝑢)ℭ(𝜆𝑣,𝑢)𝑑𝜒(𝑢) (67) gives a unique analytic extension in the complex variable𝜆∈𝖡¹for all elements𝑣∈𝔘such that _𝑚∈ℕ𝑚𝜋_𝑚∘𝑣𝔢_𝑚1²_ℂ𝑚<∞. (68)