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 Lχ2 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←χm of the probability Haar measures χm 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 subspaceℋχ2⊂Lχ2spanned 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:
(1)U(∞)=⋃{U(m):m∈ℕ},U2(∞):=U(∞)U(∞),
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=[uij]i,j∈ℕwith finitely many matrix entriesuijdistinct fromδij. We equip every group U(m) with the probability Haar measureχm.

Following [1, 2], every matrix um∈U(m) withm>1, we write in the following block matrix form:
(2)um=[zm-1abt],
corresponding to the partition m=(m-1)+1 so that zm-1∈U(m-1) and t∈ℂ. Over the group U(∞) (resp., U(m)) the right action is well defined:
(3)u·g=w-1uv,
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:
(4)πm-1m:U(m)∋um⟶um-1∈U(m-1),
such that
(5)[zm-1abt]⟼{zm-1-a(1+t)-1b:t≠-1,zm-1:t=-1,
(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 measureχm-1onU(m-1)under the mappingπm-1mis the probability Haar measureχmonU(m), that is,
(6)χm-1∘πm-1m=χm.

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 aχm-negligible set. Moreover (see [2, Lemma 3.11]), the mapping
(7)πm-1m:U′(m)⟶U′(m-1)
is continuous and surjective.

Consider the projective limits, taken with respect to the surjective Borel projectionsπm-1mand their continuous restrictionsπm-1m|U′(m), respectively,
(8)𝔘=lim⟵U(m),𝔘′=lim⟵U′(m),
called the spaces of virtual unitary matrices. Notice that𝔘is a Borel subset in the Cartesian product ⨉m∈ℕU(m)={u=(um):um∈U(m)}endowed with the product topology, because all mappingπm-1mare Borel. Moreover, the canonical projections
(9)πm:𝔘⟶U(m),πm:𝔘′⟶U′(m),
such thatπm-1=πm-1m∘πm, are surjective by surjectivity ofπm-1mandπm-1m|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),
(10)χ=lim⟵χm,
which satisfies the equalityχ=χm∘πmfor allm∈ℕ. On𝔘∖𝔘′, the measureχis zero, becauseχmis 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:
(11)πm(u·g)=w-1πm(u)v,u∈𝔘,
where m is so large that g=(v,w)∈U2(m).

The canonical dense embedding ı:U(∞)↬𝔘 to any elementum∈U(m)assigns the unique sequence u=(ul)l∈ℕ, such that
(12)ı:U(m)∋um⟼(ul)∈𝔘,ul={πll+1∘⋯∘πm-1m(um):l<m,um:l=m,[um001l-m]:l>m,
where1l-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←χm. A complex function on𝔘is called cylindrical [2, Definition 4.5] if it has the following form:
(13)f(u)=(fm∘πm)(u),u∈𝔘,
for a certainm∈ℕand a certain complex functionfmonU(m).

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

Byℒχ∞, we denote closure of the algebraic hull of all cylindricalχ-essentially bounded functions (13) with respect to the following norm:
(14)∥f∥ℒχ∞=esssupu∈𝔘|f(u)|.

Lemma 1.

The measureχ=lim←χmon𝔘is a Radon probability measure such that
(15)∫𝔘f(u·g)dχ(u)=∫𝔘f(u)dχ(u),
for allg∈U2(∞)andf∈ℒχ∞. For any compact setK⊂U(m)the following equality holds:
(16)(χ∘ı)(K)=χm(K).

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
(17)χ′=χm∘πm|𝔘′∀m∈ℕ,
if and only if for everyε>0there exists a compact set𝒦in𝔘′such that the following inequality
(18)(χm∘πm)(𝒦)≥1-ε∀m∈ℕ
holds; in this case, χ′ is uniquely determined by means of the formulaχ′(𝒦)=infm∈ℕ(χm∘πm)(𝒦), where𝒦is a compact set in𝔘′.

LetKn⊂U′(n)be a compact set with a fixed n. PuttingKn-1=πn-1n(Kn), we have
(19)χn-1(Kn-1)=(χn-1∘πn-1n)(Kn)=χn(Kn).
On the other hand, if we putKn+1=[Kn001], then via (6),
(20)χn+1(Kn+1)=(χn∘πnn+1)(Kn+1)=(χn∘πnn+1)[Kn001]=χn(Kn).
As a consequence, the compact set𝒦=(Km)in𝔘′, generated by a compact setKn⊂U′(n)with the help of mappingsπn-1n, satisfies the following condition:
(21)χn(Kn)=χm(Km)∀m∈ℕ.

The probability Haar measureχnis regular onU(n), and the complementU(n)∖U′(n)is a negligible set. Hence, ifKnruns over all compact sets inU′(n), then
(22)supKn⊂U′(n)χn(Kn)=1.
Therefore, for everyε>0there exists a compact setKn⊂U′(n)such thatχn(Kn)≥1-ε. From (21), it follows that for everyε>0the compact set𝒦satisfies the hypothesis of Prohorov’s criterion:
(23)(χm∘πm)(𝒦)=χm(Km)≥1-ε∀m∈ℕ.
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, becauseχmis zero onU(m)∖U′(m).

As a consequence of (21), we obtain (16), because
(24)χ(𝒦)=infm∈ℕχm(Km)=χn(Kn).

As is known [1, Proposition 3.2], the measureχisU2(∞)-invariant under the right actions (11) on the space𝔘. Hence, for everyf∈ℒχ∞, the equality (15) holds.

4. Shift Groups

Consider that in the spaceℒχ∞, the group of shifts
(25)Qgf(u)=f(u·g),g∈U2(∞)u∈𝔘,
is generated by the right action ofU2(∞)over𝔘. Choosing instead ofU(∞)a compact subgroup U(m)or the compact subgroups
(26)U0={g0(ϑ)=exp(𝔦ϑ):ϑ∈(-π,π]},Uj(m)={gmj(ϑ)=1j-1⊗exp(𝔦ϑ)⊗1m-j:ϑ∈(-π,π]}j=1,…,m,
we obtain the corresponding subgroups of shiftsQgwith elementsg∈U2(m)or with elements g0(ϑ)∈U02andgmj(ϑ)∈Uj2(m), respectively. Here, 1mmeans the unit element inU(m).

Lemma 2.

For anyf∈ℒχ∞the following equalities:
(27)∫𝔘fdχ=∫𝔘dχ(u)∫U2(m)Qgf(u)d(χm⊗χm)(g),(28)∫𝔘fdχ=12π∫𝔘dχ(u)∫-ππQg(ϑ)f(u)dϑ,
withg(ϑ)∈U02orUj2(m)hold.

Proof.

For anyf∈ℒχ∞, the function (u,g)↦Qgf(u)=f(u·g) is integrable on the Cartesian product 𝔘×U2(m). By the Fubini theorem, we obtain
(29)∫𝔘dχ(u)∫U2(m)Qgf(u)d(χm⊗χm)(g)=∫U2(m)d(χm⊗χm)(g)∫𝔘Qgf(u)dχ(u).

This equality yields the required formula (27), because the internal integral on the right-hand side is independent ofgand∫U2(m)d(χm⊗χm)=1. In turn, putting instead ofU(m)the subgroupsU0and Uj(m), we obtain equalities (28).

5. The Homogeneous Hilbert-Schmidt Polynomials

Consider the countable orthogonal Hilbertian sum
(30)E:=⨁m∈ℕℂm={x=(xm):xm∈ℂm,∥x∥E<∞},
with the scalar product〈x∣y〉E=∑m〈xm∣ym〉ℂm, where every coordinatexm∈ℂmis identified with its image (0,…,0,xm,0,…)∈E under the embeddingℂm↬E.

Let ⊗𝔥nE stand for the complete nth tensor power of the Hilbert subspace E, endowed with the Hilbertian scalar product and norm, respectively,
(31)〈x1⊗⋯⊗xn∣ψn〉⊗𝔥nE=∑j〈x1∣y1j〉E…〈xn∣ynj〉E,∥ψn∥⊗𝔥nE=〈ψn∣ψn〉⊗𝔥nE1/2,
where x1⊗⋯⊗xn, y1j⊗⋯⊗ynj∈⊗𝔥nE withxtj,ytj∈Efor all t=1,…,n and ψn=∑jy1j⊗⋯⊗ynj denotes a finite sum. Put ⊗𝔥0E=ℂ. We use the following short denotation:
(32)x⊗n=x⊗⋯⊗x,x∈E.

Replacing the spaceEby the subspaceℂm, we similarly define the tensor product⊗𝔥nℂm. There is the unitary embedding ⊗𝔥nℂm↬⊗𝔥nE. If m=1, then ⊗𝔥nℂ=ℂ.

For any finite sumψn=∑jy1j⊗⋯⊗ynjfrom the space⊗𝔥nℂm(or⊗𝔥nE), we can to define the finite type n-homogeneous Hilbert-Schmidt polynomials:
(33)ℂm∋x⟼〈x⊗n∣ψn〉⊗𝔥nℂm=∑j∏t=1n〈x∣ytj〉ℂm.

Consider the canonical orthonormal bases:
(34)ℰ(ℂm)={𝔢m1,…,𝔢mm}inℂm,ℰ(E)=⋃{ℰ(ℂm):m∈ℕ}inE,
where𝔢ml=(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⊙𝔥nℂmis defined to be a codomain of the symmetrization mapping:
(35)⊗𝔥nℂm∋x1⊗⋯⊗xn⟼x1⊙⋯⊙xn,x1⊙⋯⊙xn:=1n!∑𝔰∈𝔖(n)x𝔰(1)⊗⋯⊗x𝔰(n),
which is an orthogonal projector. Similarly, the symmetricnth tensor power⊙𝔥nEcan be defined. Clearly, ⊙𝔥nℂmis a closed subspace in⊙𝔥nE.

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

Let 𝔞m∈ℂmbe an arbitrary fixed element such that ∥𝔞m∥ℂm=1. Then, 𝔞m⊗n∈⊙𝔥nℂm. Using the mapping (36), we can write
(37)[πm⊗n(u)](𝔞m⊗n)=[πm(u)](𝔞m)⊗⋯⊗[πm(u)](𝔞m)︸n.
To any n-homogeneous Hilbert-Schmidt polynomial (33), there corresponds the function
(38)ψn*(u):=〈[πm⊗n(u)](𝔞m⊗n)∣ψn〉⊗𝔥nℂm=∑j∏t=1n〈[πm(u)](𝔞m)∣ytj〉ℂm
of the variableu∈𝔘. Any cylindrical function of the form𝔘∋u↦〈[πm(u)](𝔞m)∣ytj〉ℂmhas a continuous bounded restriction to𝔘′. Therefore, it isχ-essentially bounded on𝔘, because𝔘∖𝔘′ is aχ-negligible set. Consequently,ψn*∈Lχ∞andψn*|𝔘′ is continuous and bounded.

Definition 3.

We define𝒫𝔥n(ℂm)to be the space of all functionsψn*of the variableu∈𝔘, determined by the finite typen-homogeneous Hilbert-Schmidt polynomials (33).

Lemma 4.

For any element 𝔞m∈ℂm such that∥𝔞m∥ℂm=1 the set
(39)Sm={x=[πm(u)](𝔞m):u∈𝔘}
coincides with the unit sphere inℂm. As a consequence, the one-to-one antilinear corresponding
(40)⊙𝔥nℂm∋ψn⇄ψn*∈𝒫𝔥n(ℂm).
Holds, and any function ψn* is independent of the choice of an element 𝔞m∈Sm.

Proof.

Suppose, on the contrary, that there is an elementψn∈⊙𝔥nℂmsuch that〈x⊗n∣ψn〉⊗𝔥nℂm=0for all x=[πm(u)](𝔞m)∈Smwithu∈𝔘. The mapping
(41)πm:𝔘∋u⟼πm(u)∈U(m)
is surjective by surjectivity of the mappingπm(see [2, Lemma 3.1]). Hence, the set Sm coincides with the unit sphere in ℂmand is independent on the choice of an element 𝔞m. By n-homogeneity, we have 〈x⊗n∣ψn〉⊗𝔥nℂm=0 for all x∈ℂm.

Apply the following polarization formula for symmetric tensor products (see, e.g., [10, Section 1.5]):
(42)z1⊙⋯⊙zn=12nn!∑1≤t≤n∑δt=±1δ1⋯δnx⊗n,
withx=∑t=1nδtzt∈ℂm, which is valid for allz1,…,zn∈ℂm. It follows that 〈z1⊙⋯⊙zn∣ψn〉⊗𝔥nℂm=0for all elementsz1,…,zn∈ℂm. Hence, ψn=0, because the subset of all elementsz1⊙⋯⊙znis total in⊙𝔥nℂm. As a consequence, the subset
(43){x⊗n=[πm⊗n(u)](𝔞m⊗n):u∈𝔘}
is also total in⊙𝔥nℂm. It immediately yields the correspondence (40).

Consider the symmetric Fock space F and its closed subspace Fm, where
(44)F≔ℂ⊕E⊕(⊙𝔥2E)⊕(⊙𝔥3E)⊕⋯,Fm:=ℂ⊕ℂm⊕(⊙𝔥2ℂm)⊕(⊙𝔥3ℂm)⊕⋯.
We will use the following notations:
(45)(m):=(m1,…,mm),k(m):=(km1,…,kmm)∈ℤ+m,|k(m)|:=km1+⋯+kmm,k(m)!:=km1!·…·kmm!.
As is well known (see, e.g., [11]), the system of symmetric tensor elements, indexed by the set k(m),
(46)ℰ(⊙𝔥nℂm)={𝔢(m)⊗k(m)=𝔢m1⊗km1⊙⋯⊙𝔢mm⊗kmm:k(m)∈ℤ+m;|k(m)|=n𝔢(m)⊗k(m)}
forms an orthogonal basis in the subspace
(47)⊙𝔥nℂm⊂Fm.
We will also use the following notations:
(48)[m]:={(11),(21,22),…,(m1,…,mm)},{k}:={k(1),…,k(m)}∈⨉r=1mℤ+r,|{k}|:=|k(1)|+⋯+|k(m)|,{k}!:=k(1)!·…·k(m)!.
Then, the system of symmetric tensor elements with a fixedn, indexed by the sets[m]and{k},
(49)ℰn=⋃m∈ℕ{𝔢[m]⊗{k}=𝔢(1)⊗k(1)⊙⋯⊙𝔢(m)⊗k(m):𝔢(1)⊗k(1)∈ℰ(⊙𝔥|k(1)|ℂ),…,𝔢(m)⊗k(m)∈ℰ(⊙𝔥|k(m)|ℂm)withfixed|{k}|=n{𝔢[m]⊗{k}=𝔢(1)⊗k(1)⊙⋯⊙𝔢(m)⊗k(m):},
forms an orthogonal basis in the subspace⊙𝔥nE⊂F. Thus, the system
(50)ℰ={ℰn:n∈ℤ+}
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(⊙𝔥nℂm)uniquely defines the following corresponding system:
(51)ℰm,n*⊂𝒫𝔥n(ℂm),
of the following χm-integrable cylindrical functions:
(52)𝔢(m)*k(m)(u):=〈[πm⊗n(u)](𝔢m1⊗n)∣𝔢(m)⊗k(m)〉⊗𝔥nℂm=∏r=1m〈(πm∘u)(𝔢m1)∣𝔢mr〉ℂmkmr,
of the variableu∈𝔘, where we take𝔞m=𝔢m1. Consider the system of functions of the variableu∈𝔘,
(53)ℰn*=⋃m∈ℕ{𝔢[m]*{k}=𝔢(1)*k(1)·⋯·𝔢(m)*k(m):𝔢(1)*k(1)∈ℰ1,|k(1)|*,…,𝔢(m)*k(m)∈ℰm,|k(m)|*withfixed|{k}|=n{𝔢[m]*{k}=𝔢(1)*k(1)·⋯·𝔢(m)*k(m):},
generated by the system of symmetric tensor elementsℰn. All these functions belong to the spaceℒχ∞by their definition. Denote
(54)ℰ*={ℰn*:n∈ℤ+},ℰm*={ℰm,n*:n∈ℤ+}.

6. The Hardy-Type Space

LetLχ2be the space of squareχ-integrable complex functionsfon the space of virtual matrices𝔘. Since χis a probability measure, the embeddingℒχ∞⊂Lχ2holds and
(55)∥f∥Lχ2≤esssupu∈𝔘|f(u)|,f∈ℒχ∞.

Denote byℋχm2theLχ2-closure of complex linear spans of the subsystemℰm*. As is well known (see, e.g., [12, Theorem 5.6.8]), the spaceℋχm2is isomorphic to the classic Hardy spaceℋχm2(Bm)of analytic complex functions on the open unit ball Bm={xm∈ℂm:∥xm∥ℂm<1}. Therefore, the following more general definition seems natural (see, also [8]).

Definition 5.

The Hardy-type spaceℋχ2on the space of virtual unitary matrices𝔘is defined to be theLχ2-closure of the complex linear span of the systemℰ*.

Theorem 6.

The systemℰ*of all functions𝔢[m]*{k}=𝔢(1)*k(1)·⋯·𝔢(m)*k(m)withm∈ℕ, such that 𝔢(r)*k(r)∈ℰr,|k(r)|*asr=1,…,m, forms an orthogonal basis in the Hardy-type spacesℋχ2with norms
(56)∥𝔢[m]*{k}∥Lχ2=(∏r=1m(r-1)!(k)r!(r-1+|(k)r|)!)1/2.

Proof.

If|{k}|≠|{q}|, then from (28), it follows that
(57)∫𝔘𝔢[m]*{k}·𝔢-[n]*{q}dχ=∫𝔘𝔢[m]*{k}(exp(𝔦ϑ)u)·𝔢-[n]*{q}(exp(𝔦ϑ)u)dχ(u)=12π∫𝔘𝔢[m]*{k}𝔢-[n]*{q}dχ∫-ππexp(𝔦(|{k}|-|{q}|)ϑ)dϑ=0.
So, 𝔢[m]*{k}⊥𝔢[n]*{q} in the spaceLχ2if|{k}|≠|{q}|for all indices[m],[n].

Let |{k}|=|{q}| and m>n for definiteness. If the elements 𝔢[m]*{k} and 𝔢[n]*{q} 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 Qgms(ϑ) generated by elements gms(ϑ)∈Us2(m)with the subindexms,
(58)∫𝔘𝔢[m]*{k}·𝔢-[n]*{q}dχ=∫𝔘Qgms(ϑ)𝔢[m]*{k}·Qgms(ϑ)𝔢-[n]*{q}dχ=12π∫𝔘𝔢[m]*{k}·𝔢-[n]*{q}dχ∫-ππexp(𝔦kmsϑ)dϑ=0.
Hence,𝔢[m]*{k}⊥𝔢[n]*{q}inLχ2.

Let now|{k}|=|{q}|andm=n. If𝔢[m]*{k}≠𝔢[n]*{q}, then{k}≠{q}. Hence, there exists a sub-indexrsin the block-index [m]=[n] such thatkrs≠qrs. Similarly as previous mentioned, applying the formula (28) to the group of shiftsQgrs(ϑ)generated by elementsgrs(ϑ)∈Us2(r)with the sub-indexrs, we get
(59)∫𝔘𝔢[m]*{k}·𝔢-[n]*{q}dχ=12π∫𝔘𝔢[m]*{k}𝔢-[n]*{q}dχ∫-ππexp(𝔦(krs-qrs)ϑ)dϑ=0.
Hence, in this case also𝔢[m]*{k}⊥𝔢[n]*{q}under the measureχ.

Letgr=(1r,wr)∈U2(r)andu∈𝔘. Using (11) and (52), we have
(60)∫U2(r)Qgr|𝔢(r)*(k)r|2(u)d(χr⊗χr)(gr)=∫U(r)∏l=1r|〈[wr-1πr(u)](𝔢r1)∣𝔢rl〉ℂrkrl|2dχr(wr).
However, the previous integral with the Haar measureχris independent ofπr(u)∈U(r). It follows that
(61)∫U2(r)Qgr|𝔢(r)*(k)r|2(u)d(χr⊗χr)(gr)=∫U(r)∏l=1r|〈wr-1(𝔢r1)∣𝔢rl〉ℂrkrl|2dχr(wr)=(r-1)!(k)r!(r-1+|(k)r|)!=∥𝔢(r)*(k)r∥Lχr22
by the well-known formula [12, Section 1.4.9]. Using the formula (27)m-times forr=1,…,m, we get
(62)∫𝔘|𝔢[m]*{k}|2dχ=∫𝔘dχ(u)∏r=1m∫U2(r)Qgr|𝔢(r)*(k)r|2(u)d(χr⊗χr)(gr)=∏r=1m∥𝔢(r)*k(r)∥Lχr22,
because∫𝔘dχ=1. It follows that
(63)∥𝔢[m]*{k}∥Lχ22=∏r=1m∥𝔢(r)*k(r)∥Lχr22=∏r=1m(r-1)!(k)r!(r-1+|(k)r|)!,
for all𝔢[m]*{k}=𝔢(1)*k(m)·⋯·𝔢(m)*k(m).

As is known (see, e.g., [11]), the systemℰmof symmetric tensors𝔢(m)⊗(k)mwith a fixedmforms an orthogonal basis in the symmetric Fock spaceFmwith norms∥𝔢(m)⊗(k)m∥Fm=(k)m!/|(k)m|!. Similarly, the systemℰof symmetric tensors𝔢[m]⊗{k}=𝔢(1)⊗(k)1⊙⋯⊙𝔢(m)⊗k(m)with allm∈ℕ, such that𝔢(r)⊗(k)r∈ℰr,|(k)r|asr=1,…,m, forms an orthogonal basis in the symmetric Fock spaceFwith norms ∥𝔢[m]⊗{k}∥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
(64)𝔢(m)⊗(k)m∥𝔢(m)⊗(k)m∥Fm⇄𝔢(m)*(k)m∥𝔢(m)*(k)m∥Lχm2,𝔢[m]⊗{k}∥𝔢[m]⊗{k}∥F⇄𝔢[m]*{k}∥𝔢[m]*{k}∥Lχ2,
uniquely define the corresponding anti-linear isometric isomorphisms
(65)Fm≃ℋχm2(Bm),F≃ℋχ2.

Reasoning by analogy with [8, Proposition 6.1 and Theorem 7.1], it is easy to show that the Hardy space ℋχ2possesses the reproducing kernel of a Cauchy type
(66)ℭ(v,u)=∑n∈ℤ+∑|{k}|=n𝔢[m]*{k}(v)𝔢-[m]*{k}(u)∥𝔢[m]*{k}∥Lχ22=∏m=1∞(1-〈(πm∘v)(𝔢m1)|(πm∘u)(𝔢m1)〉E)-m,
withu,v∈𝔘, where the sum∑|{k}|=nis over all indices{k}∈{⨉r=1mℤ+r:m∈ℕ}such that |{k}|=n. As a consequence, the integral representation of any functionf∈ℋχ2,
(67)f(λv)=∫𝔘f(u)ℭ(λv,u)dχ(u)
gives a unique analytic extension in the complex variableλ∈B1for all elementsv∈𝔘such that
(68)∑m∈ℕm∥(πm∘v)(𝔢m1)∥ℂm2<∞.

NeretinY. A.Hua-type integrals over unitary groups and over projective limits of unitary groupsOlshanskiG.The problem of harmonic analysis on the infinite-dimensional unitary groupPickrellD.Measures on infinite dimensional Grassmann manifoldsKerovS.OlshanskiG.VershikA.Harmonic analysis on the infinite symmetric group: a deformation of regular representationBorodinA.OlshanskiG.Harmonic analysis on the infinite-dimensional unitary group and determinantal point processesTomasE.On Prohorov's criterion for projective limitsYamasakiY.Projective limit of Haar measures on O(n)LopushanskyO.ZagorodnyukA.Hardy type spaces associated with compact unitary groupsBourbakiN.FloretK.Natural norms on symmetric tensor products of normed spacesReedM.SimonB.RudinW.