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

Authors: Oleh Lopushansky
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction

We investigate an orthogonal system of the Hilbert-Schmidt polynomials in the spaceLχ2 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 Theorems6–7 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.LetU′(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 embeddingı: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).Let𝔞m∈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 element𝔞m∈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 spaceF 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 Lemma4, 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<∞.



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*Source: 810735-2013-07-14.xml*