paley-wiener isomorphism over infinite-dimensional unitary...

Results Math 72 (2017), 2101–2120c© 2017 The Author(s). This article is an open accesspublication1422-6383/17/042101-20published online September 20, 2017DOI 10.1007/s00025-017-0750-0 Results in Mathematics

Paley-Wiener Isomorphism OverInfinite-Dimensional Unitary Groups

Oleh Lopushansky

Abstract. An analog of the Paley-Wiener isomorphism for the Hardy spacewith an invariant measure over infinite-dimensional unitary groups is de-scribed. This allows us to investigate on such space the shift and mul-tiplicative groups, as well as, their generators and intertwining opera-tors. We show applications to the Gauss-Weierstrass semigroups and tothe Weyl–Schrodinger irreducible representations of complexified infinite-dimensional Heisenberg groups.

Mathematics Subject Classification. 46T12, 46G20.

Keywords. Hardy spaces of infinitely many variables, Harmonic analysison infinite-dimensional groups, Symmetric Fock spaces.

1. Introduction

The work deals with the Hardy space H2χ of square-integrable C-valued func-

tions with respect to a probability measure χ over the infinite-dimensionalunitary group U(∞) :=

⋃{U(m) : m ∈ N}, extended by unit 1, which irre-

ducibly acts on a separable complex Hilbert space E with an orthonormalbasis {em}. Here, U(m) is the subgroup of unitary m × m-matrices endowedwith Haar’s measure χm.

In what follows, U(∞) is densely embedded via a universal mapping πinto the space of virtual unitary matrices U = lim←−U(m) defined as the projec-tive limit under Livsic’s mappings πm+1

m : U(m + 1) → U(m). The projectivelimit χ = lim←−χm, such that each image-measure πm+1

m (χm+1) is equal to χm,is concentrated on the range π(U(∞)) consisting of stabilized sequences (see[18,20]). The measure χ is invariant under right actions [20, n.4].

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2102 O. Lopushansky Results Math

We refer to [5,26] for applications of χ to stochastic processes. Neededproperties of Hardy spaces H2

χ can be found in [15]. Various cases of Hardyspaces in infinite-dimensional settings were considered in [9,17].

Now, we briefly describe results. Using a unitarily weighted symmetricFock space (Γw, 〈· | ·〉w) with a canonical orthogonal basis of symmetric tensorproducts {e�λ

ı } of basis elements {em} ⊂ E indexed by Young diagrams λ andnormalized by measure χ, we find an orthogonal basis of polynomial {φλ

ı } inH2

χ such that the conjugate-linear mapping

Φ : Γw → H2χ

is a surjective isometry with one-to-one correspondence e�λı � φλ

ı . This allowsus to establish in Theorem 2 an integral formula for a Fock-symmetric F-transform

F : H2χ f → f ∈ H2

w

where the Hilbert space H2w, uniquely determined by Γw, consists of Hilbert–

Schmidt analytic entire functions on E. Thus, the F-transform acts as ananalog of the Paley-Wiener isomorphism over infinite-dimensional groups.

Furthermore, we investigate two different representations of the additivegroup (E,+) over the Hardy space H2

χ by shift and multiplicative groups.Theorem 3 states that the F-transform is an intertwining operator betweenthe multiplication group M†

a on H2χ and the shift group Ta on H2

w. On theother hand, Theorem 4 shows that F is the same between the shift group T †

a

on H2χ and the multiplication group Ma∗ on H2

w. Integral formulas describinginterrelations between their generators are established. In Theorem 5 suitablecommutation relations are stated.

Applications to the Gauss-Weierstrass-type semigroups on H2χ are shown

in Theorem 6. Another application to linear representations of complexifiedinfinite-dimensional Heisenberg groups on H2

χ in a Weyl–Schrodinger form isgiven in Theorem 7.

Infinite-dimensional Heisenberg groups was considered in [16] by usingreproducing kernel Hilbert spaces. The Schrodinger representation of infinite-dimensional Heisenberg groups on L2

γ with respect to a Gaussian measure γover a real Hilbert space is described in [3] (see also earlier publications [1,2]).

In conclusion, we note that a motivation for this study was the follow-ing simple relations in the Hardy space H2

χ over 1-dimensional group U(1) ={u = exp(iϑ) : ϑ ∈ [−π, π]}. In this case, {un : n ∈ Z+} is an orthonormal ba-sis and Γw � 2, since Φ∗f = (fn) ∈ 2 for any f ∈ H2

χ with Fourier coeffi-cients fn =

∫f(u)undu = 〈un | Φ∗f〉w. While, f(x) =

∫f(u) exp(xu) du =∑

fnxn/n! = 〈ε(x) | Φ∗f〉w, where ε(x) = (xn/n!) ∈ 2 for all x ∈ C.Moreover, the equalities Taf(x) =

∫f(u) exp[(x + a)u] du = F(M†

af)(x)with M†

af(u) = exp(au)f(u) hold for all x, a ∈ C. On the other hand, (Maf)(x)= exp(xa)

∑fnxn/n! =

∑(T †

afn)xn/n! with T †afn =

∑nk=0

(nk

)akfn−k. Hence,

Vol. 72 (2017) Paley-Wiener Isomorphism 2103

Maf = F(T †af), where (T †

af)(u) =∑

(T †afn)un. In result, TaMb = exp(ab)

MbTa for all a, b ∈ C and the Weyl–Schrodinger representation of Heisenberg’sgroup from Theorem 7 retains a classic form.

The case H2χ over m-dimensional group U(m) is similar with a proviso

that the weighted Fock space Γw is normalized by ‖e�λı ‖w =

(n+m−1

n

)−1/2

where n = |λ| is a homogeneity degree of the basis polynomial φλı in H2

χ.Note that the normalization ‖e�λ

ı ‖w = n!−1/2 with n = |λ| leads to the caseof Segal-Bargmann’s space H2

γ with standard centered probability Gaussianmeasure γ on C

m.

2. Hilbert–Schmidt Analyticity

Let E stand for a separable complex Hilbert space with scalar product 〈· | ·〉norm ‖ · ‖ and a fixed orthonormal basis {ek : k ∈ N}. Denote by E⊗n

alg =

E⊗ n times· · · ⊗E (n ∈ N) its algebraic tensor power consisted of the linearspan of elements ψn = x1 ⊗ · · · ⊗ xn with xi ∈ E (i = 1, . . . , n). Set x⊗n :=

x⊗ n times· · · ⊗x. The symmetric algebraic tensor power E�nalg = E � · · · � E is

defined to be the range of the projector sn : E⊗nalg ψn → x1 � · · · � xn with

x1 � · · · � xn := (n!)−1∑

σxσ(1) ⊗ · · · ⊗ xσ(n) where σ : {1, . . . , n} → {σ(1),. . . , σ(n)} runs through all permutations. The symmetric algebraic Fock spaceis defined as the algebraic direct sum Γalg =

∑n∈Z+

E�nalg with E�0

alg = C.

Let E⊗nh := E ⊗h · · · ⊗h E be the completion of E⊗n

alg by Hilbertian norm

‖ψn‖h = 〈ψn | ψn〉1/2h with 〈ψn | ψ′

n〉h = 〈x1 | x′1〉 · · · 〈xn | x′

n〉. Denote by E�nh

the range of continuous extension of sn on E⊗nh . As usual, the symmetric Fock

space is defined to be the Hilbertian direct sum Γh =⊕

n∈Z+E�n

h .Denote by λ = (λ1, . . . , λm) ∈ Z

m+ with λ1 ≥ λ2 ≥ · · · ≥ λm a partition

of n ∈ N, that is, n = |λ| where |λ| := λ1 + · · · + λm. Any λ may be identifiedwith Young’s diagram of length l(λ) = m. Let Y denote all diagrams andYn = {λ ∈ Y : |λ| = n}. Assume that Y0 = {∅ ∈ Y : |∅| = 0} and l(∅) = 1. LetN

m∗ := {ı = (ı1, . . . , ım) ∈ N

m : ıl =/ ık, ∀ l =/ k}. For each λ ∈ Y we assign theconstant

C|λ|,l(λ) :=(l(λ) − 1)!|λ|!

(l(λ) − 1 + |λ|)! ≤ 1. (1)

The spaces E�nalg and Γalg may be generated by the basis of symmetric

tensors

e�Yn =⋃{

e�λı := e⊗λ1

ı1 � · · · � e⊗λl(λ)ıl(λ) : (λ, ı) ∈ Yn × N

l(λ)∗

},

e�Y =⋃{

e�Yn : n ∈ Z+

}with e�∅

ı = 1,


respectively. As is known [4, Sect. 2.2.2], norm of basis element in Γh is equalto

‖e�λı ‖2h =

λ!|λ|! , λ! := λ1! · · · λm!. (2)

Let us define a new Hilbertian norm on Γalg by the equality‖ · ‖w = 〈· | ·〉1/2

w where scalar product 〈· | ·〉w is determined via the orthogonalrelations

〈e�λı | e�λ′

ı′ 〉w ={

C|λ|,l(λ)‖e�λı ‖2h : λ = λ and ı = ı′,

0 : λ =/ λ′ or ı =/ ı′.

Denote by E�nw and Γw the appropriate completions of E�n

alg and Γalg, respec-

tively. For any ı ∈ Nl(λ)∗ there corresponds in E�n

w the d-dimension subspacewith d = C−1

|λ|,l(λ), spanned by elements{e�λı : λ ∈ Yn

}. The Hilbertian or-

thogonal sum

Γw =⊕

n∈Z+

E�nw

endowed with 〈· | ·〉w we will call unitarily weighted symmetric Fock space.Let x =

∑ekxk be the Fourier series of x ∈ E with coefficients xk =

〈x | ek〉. We assign to any (λ, ı) ∈ Yn × Nl(λ)∗ the n-homogenous Hilbert–

Schmidt polynomial defined via the Fourier coefficients

xλı := 〈x⊗n | e�λ

ı 〉w = xλ1ı1 . . . x

λl(λ)ıl(λ) , x ∈ E.

Using the tensor multinomial theorem, we define in Γw the Fourier decompo-sition of exponential vectors (or coherent state vectors)

ε(x) :=⊕

n∈Z+

x⊗n

n!=

⊕

n∈Z+

1n!

(∑

k∈N

ekxk

)⊗n

=⊕

n∈Z+

1n!

∑

(λ,ı)∈Yn×Nl(λ)∗

n!λ!

e�λı xλ

ı

(3)

with respect to the basis e�Y. It is convergent in Γw in view of (1) and

‖ε(x)‖2w =∑

n∈Z+

1n!2

∑


(n!λ!

)2‖e�λ

ı ‖2w|xλı |2

=∑

n∈Z+

1n!2

∑

(λ,ı)

n!λ!

C|λ|,l(λ)|xλı |2 ≤

∑

n∈Z+

1n!

∑

(λ,ı)

n!λ!

|xλı |2

=∑

n∈Z+

1n!

(∑

k∈N

|xk|2)n

= e‖x‖2.

(4)

Particulary, (4) implies that the function E x → ε(x) ∈ Γw is entire analytic.


Definition 1. The space of C-valued Hilbert–Schmidt entire analytic functionsin variable x ∈ E, associating with the unitarily weighted symmetric Fockspace Γw, is defined to be

H2w := {ψ∗(x) := 〈ε(x) | ψ〉w : ψ ∈ Γw} with the norm ‖ψ∗‖ = ‖ψ‖w .

Every function ψ∗ is entire analytic as the composition of ε(·) with〈· | ψ〉w. The subspace in H2

w of n-homogenous Hilbert–Schmidt polynomialsis defined to be

H2,nw =

{ψ∗

n(x) = 〈x⊗n | ψn〉w : ψn ∈ E�nw

}.

Evidently, H2w = C ⊕ H2,1

w ⊕ H2,2w ⊕ . . ..

It is important that H2w is uniquely determined by Γw since {ε(x) : x ∈ E}

is total in Γw. Similarly, for the subspace H2,nw which is uniquely determined

by E�nw , since {x⊗n : x ∈ E} is total in E�n

w . The last totality follows from thepolarization formula for symmetric tensor products

e�λı =

12nn!

∑

θ1,...,θn=±1

θ1 . . . θn a⊗n with a =l(λ)∑

i=1

θie⊗λiıi

(5)

which is valid for all e�λı ∈ e�Yn (see e.g. [11, Sect. 1.5]) Thus, the conjugate-

linear isometries ψ → ψ∗ from Γw onto H2w and from E�n

w onto H2,nw hold.

In conclusion, we can notice that every analytic function ψ∗ ∈ H2w deter-

mined by ψ =⊕

ψn ∈ Γw, (ψn ∈ E�nw ) has the Taylor expansion at zero

ψ∗(x) =∑

n∈Z+

1n!

∑


〈e�λı | ψn〉w‖e�λ

ı ‖2wxλ

ı , x ∈ E

that follows from (3). The function ψ∗ is entire Hilbert–Schmidt analytic [15,n.5].

Note that analytic functions of Hilbert–Schmidt types were also consid-ered in [10,14,21]. More general classes of analytic functions associated withcoherent sequences of polynomial ideals were described in [8].

3. Hardy Space Over U(∞)

In what follows, we endow each group U(m) with the probability Haar mea-sure χm and assume that U(m) is identified with its range with respect to

the embedding U(m) um →[um 00 1

]

∈ U(∞). The Livsic transform from

U(m + 1) onto U(m) is described in [18, Proposition 0.1] and [20, Lemma 3.1]as the surjective Borel mapping

πm+1m : um+1 :=

[zm ab t

]

−→ um :={

zm − [a(1 + t)−1b] : t �= −1zm : t = −1.


The projective limit U := lim←−U(m) under πm+1m has surjective Borel projec-

tions πm : U u → um ∈ U(m) such that πm = πm+1m ◦ πm+1.

Consider a universal dense embedding π : U(∞) � U which to everyum ∈ U(m) assigns the stabilized sequence u = (uk) such that (see [20, n.4])

π : U(m) um → (uk) ∈ U, uk ={

πmk (um) : k < m

um : k ≥ m,(6)

where πmk := πk+1

k ◦ . . . ◦ πmm−1 for k < m and πm

k is identity mapping fork ≥ m. On its range π(U(∞)), endowed with the Borel structure from U, weconsider the inverse mapping

π−1 : Uπ → U(∞) where Uπ := π(U(∞)).

The right action Uπ u → u.g ∈ Uπ with g = (v, w) ∈ U(∞) × U(∞) isdefined by πm(u.g) = w−1πm(u)v where m is so large that g = (v, w) ∈U(m) × U(m).

Following [18, n.3.1], [20, Lemma 4.8] via the Kolmogorov consistencytheorem (see e.g. [19, Theorem 1], [24, Corollary 4.2]) we uniquely define onU = lim←−U(m) the probability measure χ := lim←−χm such that each image-measure πm+1

m (χm+1) is equal to χm. For any Borel subset A ⊂ Uπ we haveπm+1(A) ⊆ (πm+1

m )−1 [πm(A)], because πm = πm+1m ◦ πm+1. It follows that

(χm ◦ πm)(A) = πm+1m (χm+1)[πm(A)] = χm+1[(πm+1

m )−1[πm(A)]] ≥ (χm+1 ◦πm+1)(A). Hence, χ satisfies the condition

χ(A) = inf(χm ◦ πm)(A) = lim χm(A) (7)

and therefore the projective limit lim←−χm exists on Uπ via the well knownProhorov theorem [6, Theorem IX.52]. Moreover, it is a Radon probabilitymeasure concentrated on Uπ [24, Theorem 4.1]. By the known portmanteautheorem [13, Theorem 13.16] and Fubini’s theorem the invariance of Haarmeasures χm together with (7) yield the following invariance properties underthe right action

∫

f(u.g) dχ(u) =∫

f(u) dχ(u), g ∈ U(∞) × U(∞), f ∈ L∞γ ,

∫

f dχ =∫

dχ(u)∫

U(m)×U(m)

f(u.g) d(χm ⊗ χm)(g),

where L∞χ stands for the space of all χ-essentially bounded complex-valued

functions defined on Uπ and endowed with norm ‖f‖∞ = ess supu∈Uπ|f(u)|.

Let L2χ be the space of square-integrable C-valued functions f on Uπ with

norm

‖f‖χ = 〈f | f〉1/2χ where 〈f | f〉χ :=

∫

f1f2 dχ.

The embedding L∞χ � L2

χ holds, moreover, ‖f‖χ ≤ ‖f‖∞ for all f ∈ L∞χ .


To given the E-valued mapping Uπ u → π−1(u)e1, we can well-definethe Borel χ-essentially bounded functions in the variable u ∈ Uπ,

φk := φek, φek

(u) =⟨π−1(u)e1 | ek

⟩, k ∈ N,

which do not depend on the choice of e1 in⋃

S(m) where S(m) isthe m-dimensional unit sphere in E [15, n.3]. The uniqueness ofφx(u) = 〈π−1(u)e1 | x〉 with x ∈ E results from the total embeddingπ : U(∞) � U. From (6) it follows that π−1 ◦ π−1

m coincides with the em-bedding U(m) � U(∞). Hence, by (7) and the portmanteau theorem thereexist the limit

∫

φx dχ = limm→∞

∫

U(m)

φx d(χm ◦ πm) = limm→∞

∫

U(m)

(φx ◦ π−1m ) dχm,

i.e., φx ∈ L∞χ for any φx(u) = 〈π−1(u)e1 | x〉 with x ∈ E.

By formula (5) to every e�λı ∈ e�Yn there uniquely corresponds the Borel

function from L∞χ

φλı (u) :=

⟨[π−1(u)e1]⊗n | e�λ

ı

⟩w

= φλ1ı1 (u) . . . φ

λl(λ)ıl(λ) (u)

in the variable u ∈ Uπ. It follows that the orthogonal basis e�Y of elementse�λı = e⊗λ1

ı1 � · · · � e⊗λmım

, indexed by λ = (λ1, . . . , λm) ∈ Y and ı = (ı1, . . . , ım)∈ N

m∗ with m = l(λ), uniquely determines the systems of Borel χ-essentially

bounded functions in the variable u ∈ Uπ,

φYn =⋃{

φλı := φλ1

ı1 · · · φλmım

: (λ, ı) ∈ Yn × Nm∗ , m = l(λ)

},

φY =⋃{

φYn : n ∈ Z+

}with φ∅

ı ≡ 1.

Definition 2. The Hardy space H2χ is defined as the closed complex linear span

of φY endowed with L2χ-norm.

The following assertion is proved in [15, Theorem 3.2].

Theorem 1. The system of Borel functions φY forms an orthogonal basis inH2

χ such that

‖φλı ‖χ = C

1/2|λ|,l(λ)‖e�λ

ı ‖h, λ ∈ Y, ı ∈ Nl(λ)∗ .

Define the subspace H2,nχ ⊂ H2

χ for any n ∈ N to be the closed linearspan of the subsystem φYn . Theorem 1 implies that H2,n

χ ⊥ H2,mχ in L2

χ forany n =/ m. This provides the orthogonal decomposition

H2χ = C ⊕ H2,1

χ ⊕ H2,2χ ⊕ · · · .


4. Fock-Symmetric F -Transform

The one-to-one correspondence e�λı � φλ

ı allows us to define via the changeof orthonormal bases

Φ : Γw e�λı ‖e�λ

ı ‖−1w → φλ

ı ‖φλı ‖−1

χ ∈ H2χ, λ ∈ Y, ı ∈ N

l(λ)∗

the isometric conjugate-linear mapping Φ : Γw → H2χ. The adjoint mapping

Φ∗ : H2χ → Γw is defined by

⟨Φe�λ

ı | f⟩

χ=⟨e�λı | Φ∗f

⟩w

with f ∈ H2χ. The

suitable Fourier decomposition has the form

Φψ =∑

(λ,ı)∈Y×Nl(λ)∗

ψ(λ,ı)φλı ‖φλ

ı ‖−1χ , ψ(λ,ı) := 〈e�λ

ı | ψ〉w ‖e�λı ‖−1

w

for any ψ ∈ Γw. In particular, the equality Φx =∑

xkφk is valid for all x ∈ E.This gives the equalities

‖Φx‖2χ =∑

|xk|2 = ‖x‖2, x ∈ E.

Using this, we will examine the composition of Φ with the Γw-valued functionε : E x → ε(x). Its correctness justifies the following assertion that substan-tially uses the L∞

χ -valued function

φx : Uπ u → (Φx)(u) =∑

xkφk(u)

which is linear in the variable x ∈ E.Similarly to the known case of Wiener spaces, the function Φx can be

seen as a group analog of the Paley-Wiener map (see e.g. [12, n.4.4] or [23]).

Lemma 1. The composition Φε(x), which is understood as the function

[Φε(x)](u) : Uπ u → exp (φx(u)) ,

takes values in L∞χ for all x ∈ E.

Proof. Applying Φ to the Fourier decomposition (3), we obtain

Φε(x) =∑

n∈Z+

1n!

∑


n!λ!

xλı φλ

ı =∑

n∈Z+

1n!

(∑

k∈N

xkφk

)n

= exp (φx) .

It directly follows that ‖Φε(x)‖∞ ≤ exp ‖φx‖∞. �

Theorem 2. For every f =∑

fn ∈ H2χ, (fn ∈ H2,n

χ ) the entire analytic func-tion f(x) := 〈ε(x) | Φ∗f〉w in the variable x ∈ E and its Taylor coefficients atorigin have the integral representations

f(x) =∫

exp(φx)f dχ and dn0 f(x) =

∫

φnxfn dχ, (8)

respectively. The mapping F : H2χ f → f ∈ H2

w (regarded as a Fock-symmet-ric F-transform) provides the isometries

H2χ � H2

w and H2,nχ � H2,n

w .


Proof. First recall that the Γw-valued function ε(·) is entire analytic on E,therefore f is the same, as the composition of ε(·) with 〈· | Φ∗f〉w. Farther on,consider the Fourier decomposition with respect to the basis φY,

f =∑

n∈Z+

fn =∑

n∈Z+


fλ,ı,nφλı

‖φλı ‖χ

, fλ,ı,n =1

‖φλı ‖χ

∫

f φλı dχ.

Applying Φ∗ to f in this decomposition and substituting fλ,ı,n into f , weobtain

f(x) =∑

n∈Z+

1n!

∑


n!λ!

fλ,ı,n〈e�λı | e�λ

ı 〉wxλı

‖e�λı ‖w

=∫ ∑

n∈Z+

1n!

( ∑


n!λ!

xλı φλ

ı

)f dχ =

∫

exp(φx

)f dχ

where the last equality is valid by Lemma 1. It particularly follows that fory = αx,

f (y) =∫

exp(φαx

)f dχ =

∑αn

∫φn

x

n!fndχ, α ∈ C.

Differentiating f at y = 0 and using the n-homogeneity of derivatives, weobtain

dn0 f(x) =

dn

dαn

∑αn

∫φn

x

n!fn dχ

∣∣∣α=0

=∫

φnxfn dχ.

Finally, we notice that the isometry H2χ � H2

w holds, since the isometry Φ∗ issurjective. In the case of polynomials we similarly get H2,n

χ � H2,nw . �

Note that a different integral formula for analytic functions employingWiener measures on infinite-dimensional Banach spaces was presented in [22].

5. Exponential Creation and Annihilation Groups

Let us define the linear mapping jn : E�nw → E�n

h to be the continuous ex-tension of identity mapping acting on the dense subspace E�n

alg ⊂ E�nw ∩ E�n

h .Such continuous extension jn is a contractive injection with dense range. Infact, it suffices to expand elements from E�n

w and E�nh into the Fourier series

with respect to orthogonal basis e�Yn and apply the inequality

‖e�λı ‖2w = C|λ|,l(λ)‖e�λ

ı ‖2h ≤ ‖e�λı ‖2h, λ ∈ Yn (9)

which follows from Theorem 1, taking into account the inequality (1). Us-ing subsequently that E�n

h is reflexive, we obtain that its adjoint operator


j∗n : E�nh → E�n

w is a contractive injection with dense range. Thus, the map-

ping jn is also injective. Moreover, E�nh

j∗n→ E�nw

jn� E�nh forms a Gelfand triple.

Particularly, the operator sn possesses continuous extension on E�nw .

Using this, we consider the linear operator

sn/m := sn ◦ (jm ⊗ jn−m) with m ≤ n

defined to be φm � ψn−m = sn/m(φm ⊗ ψn−m) ∈ E�nw for all φm ∈ E�m

w ,ψn−m ∈ E

�(n−m)w .

Lemma 2. The mapping sn/m from E�mw ⊗h E

�(n−m)w to E�n

w is a contractiveinjection with dense range.

Proof. Expand elements of E�mw ⊗h E

�(n−m)w with respect to e�λ

ı ⊗ e�μj for

all λ, μ ∈ Y, ı ∈ Nl(λ), j ∈ N

l(μ) such that |λ| = m, |μ| = n − m. Using (9), wehave

‖e�λı ⊗ e�μ

j ‖E�m

w ⊗hE�(n−m)w

= ‖e�λı ‖w‖e�μ

j ‖w≤ ‖e�λ

ı ‖h‖e�μj ‖h = ‖e�λ

ı ⊗ e�μj ‖h.

As above, it implies that the mapping jm ⊗ jn−m : E�mw ⊗h E

�(n−m)w → E⊗n

h ,defined to be the continuous extension of identity mapping on E�m

alg ⊗E�(n−m)alg ,

is a contractive injection. Using subsequently that E�mh ⊗h E

�(n−m)h is reflex-

ive, we get the Gelfand triple

E�mw ⊗h E�(n−m)

w

sn/m→ E�nh

j∗n→ E�nw

where injections are contractive and have dense ranges. �

Lemma 3. The Γw-valued function, defined on {ε(x) : x ∈ E} by

Taε(x) = ε(x + a),

has a unique linear extension Ta : Γw ψ → Taψ ∈ Γw such that

‖Taψ‖2w ≤ exp(‖a‖2)‖ψ‖2w and Ta+b = TaTb = TbTa for all a, b ∈ E.

Proof. Let us define the creation operators δma,n : E

�(n−m)w → E�n

w (m ≤ n) as

δma,nx⊗(n−m) := sn/m

[a⊗m ⊗ x⊗(n−m)

]=

(n − m)!n!

dm(x + ta)⊗n

dtm

∣∣∣t=0

(10)

for all a, x ∈ E. Note that the second equality in (10) follows from the bino-mial formula for symmetric tensor elements (x + ta)⊗n =

∑nm=0

(nm

)(ta)⊗m �

x⊗(n−m). Put δ0a,n = 1. If a = 0 then δm0,n = 0. Summing over n ≥ m with

coefficients 1/(n − m)!, we get

δma ε(x) =

dmε(x + ta)dtm

∣∣∣t=0

=⊕

n≥m

sn/m[a⊗m ⊗ x⊗(n−m)](n − m)!

, t ∈ C. (11)


This series is convergent, since by Lemma 2 and (4) the inequality

‖δma ε(x)‖w ≤ ‖a‖m

∥∥∥⊕

n≥m

x⊗(n−m)

(n − m)!

∥∥∥w

= ‖a‖m ‖ε(x)‖w (12)

holds. From (11) and the tensor binomial formula mentioned above it followsthat

n⊕

m=0

1m!

δma,n

x⊗(n−m)

(n − m)!=

n⊕

m=0

a⊗m � x⊗(n−m)

m!(n − m)!=

(x + a)⊗n

n!.

Summing over n ∈ Z+ with coefficients 1/n! and using (11), we obtain

Taε(x) =⊕

n∈Z+

n∑

m=0

1m!

δma,n

x⊗(n−m)

(n − m)!

=∑

m∈Z+

1m!

⊕

n≥m

δma,n

x⊗(n−m)

(n − m)!= exp(δa)ε(x).

The inequalities (4) and (12) yield ‖Taε(x)‖2w ≤ exp(‖a‖2

)‖ε(x)‖2w. Taking

into account the totality of {ε(x) : x ∈ E}, this inequality implies the requiredinequality on Γw. It also follows that Ta+b = TaTb = TbTa, since δa+b = δa + δbfor all a, b ∈ E by linearity of creation operators. This ends the proof. �

We define the adjoint operators δ∗ma,n : E�n

w ψn → δ∗ma,nψn ∈ E

�(n−m)w as

⟨δma,nx⊗(n−m) | ψn

⟩w

=⟨x⊗(n−m) | δ∗m

a,nψn

⟩w, a, x ∈ E

for n ≥ m. It immediately follows that for every ψn−m ∈ E�(n−m)w and x ∈ E,

⟨δ∗ma,nx⊗n | ψn−m

⟩w

=⟨x⊗n | δm

a,nψn−m

⟩w

=⟨x⊗n | a⊗m � ψn−m

⟩w

= 〈x | a〉m⟨x⊗(n−m) | ψn−m

⟩w.

(13)

Using δ∗ma,n , we can uniquely define a Γw-valued function T ∗

a by the equalities

T ∗a ε(x) = exp(δ∗

a )ε(x) =∑

m∈Z+

δ∗ma ε(x)

m!, δ∗m

a ε(x) :=⊕

n≥m

δ∗ma,nx⊗n

n!(14)

for all a, x ∈ E. Taking into account Lemma 3, we obtain the following claim.

Lemma 4. The Γw-valued function T ∗a , defined by (14), possesses a unique

linear extension T ∗a : Γw ψ → T ∗

a ψ ∈ Γw such that

‖T ∗a ψ‖2w ≤ exp(‖a‖2) ‖ψ‖2w and T ∗

a+b = T ∗a T ∗

b = T ∗b T ∗

a for all a, b ∈ E.

Definition 3. We will call the Γw-valued functions Ta and T ∗a in variable a ∈ E

the exponential creation and annihilation groups, respectively.


6. Intertwining Properties of F -Transform

Let us define on the space H2χ the multiplicative group M†

a : E a → M†a to

be

M†a f(u) = exp[φa(u)]f(u), f ∈ H2

χ, u ∈ Uπ.

It can be considered as a linear representation of the additive group (E,+).By Lemma 1 the function u → exp[φa(u)] with a fixed a belongs to L∞

χ . Hence,M†

a is continuous on H2χ. The generator of the 1-parameter group C t → M†

ta

coincides with the operator of multiplication by the L∞χ -valued function

φa : Uπ u → φa(u) where dM†ta/dt|t=0 = φa.

The continuity of E a → exp(φa) implies that this 1-parameter groupM†

ta is strongly continuous on H2χ. As a consequnce, its generator (φaf)(u) =

φa(u)f(u) with domain D(φa) ={f ∈ H2

χ : φaf ∈ H2χ

}is closed and densely-

defined. As well, its power φma defined on D(φm

a ) ={f ∈ H2

χ : φma f ∈ H2

χ

}for

any m ∈ N is the same (see, e.g. [7] for details).The additive group (E,+) may be also linearly represented on H2

w as theshift group

Taf(x) = f(x + a), f ∈ H2χ, x, a ∈ E.

The directional derivative on the space H2w along a nonzero a ∈ E coincides

with the generator of the 1-parameter shift subgroup C t → Tta, that is,

daf = limt→0

t−1(Ttaf − f) with domain D(da) :={f ∈ H2

w : daf ∈ H2w

}.

Note that the 1-parameter shift group Tta, which is intertwined with M†ta by

the F-transform

Ttaf(x) =∫

exp[φx+ta

]f dχ =

∫

exp(φx)M†taf dχ, (15)

is strongly continuous on H2w. Since D(dm

a ) contains all polynomials from H2w,

each operator dma with domain D(dm

a ) ={f ∈ H2

w : dma f ∈ H2

w

}is closed and

densely-defined. From (15) it directly follows

dma f(x) =

∫

exp(φx)dmM†

ta

dtm

∣∣∣t=0

f dχ =∫

exp(φx)φma f dχ (16)

for all f ∈ D(φma ) and x ∈ E. On the other hand, by Theorem 2 we have

Taf(x) = 〈Taε(x) | Φ∗f〉 = 〈ε(x) | T ∗a Φ∗f〉 =

∫

exp(φx)ΦT ∗a Φ∗f dχ. (17)

Theorem 2 together with (15) and (17) imply that M†a is connected with the

exponential annihilation group T ∗a by the intertwining operator Φ. This can

be written as M†a = ΦT ∗

a Φ∗. Thus, the F-transform serves as an intertwining


operator for the groups M†a on H2

χ. Moreover, using (15), (16) and (17), weobtain

dmTtaf(x)/dtm|t=0 =⟨ε(x) | δ∗m

a Φ∗f⟩w

= dma f(x).

As a result, we have proved the following statement.

Theorem 3. For every f ∈ H2χ the following equalities hold,

TaF(f) = F(M†a f), M†

a f = ΦT ∗a Φ∗f, a ∈ E,

Moreover, for every f ∈ D(φma ) (m ∈ N) and a nonzero a ∈ E,

dma f(x) = 〈ε(x) | δ∗m

a Φ∗f〉w =∫

exp(φx)φma f dχ, x ∈ E.

Let us consider on H2w the multiplicative group with a nonzero a ∈ E,

Ma∗ f(x) = f(x) exp〈x | a〉, f ∈ H2w.

The generator on H2w of the appropriate 1-parameter subgroup C t → Mta∗

is

dMta∗/dt|t=0 = 〈· | a〉 := a∗, a ∈ E.

Hence, it coincides with the following linear operator of multiplication

(a∗f)(x) = 〈x | a〉 f(x) with domain D(a∗) ={f ∈ H2

w : a∗f ∈ H2w

}.

Its power a∗m is densely-defined on D(a∗m) ={f ∈ H2

w : a∗mf ∈ H2w

}which

contains all polynomials from H2w.

Using Lemma 3 we can represent the additive group (E,+) over the spaceH2

χ by the shift group

T †a = ΦTaΦ

∗ with the generator δ†a = ΦδaΦ

∗

defined on D(δ†a) =

{f ∈ H2

χ : δ†af ∈ H2

χ

}This means that T †

a is connected viathe intertwining operator Φ with the exponential creation group Ta.

Theorem 4. For every f ∈ H2χ the following equality holds,

Ma∗F(f) = F(T †a f), a ∈ E,

that is, the F-transform is an intertwining operator for the groups Ma∗ onH2

w and T †a on H2

χ. Moreover, for every f ∈ D(δ†ma ) =

{f ∈ H2

χ : δ†ma f ∈ H2

χ

}

(m ∈ N) and a nonzero a ∈ E,

(a∗mf)(x) = 〈ε(x) | δma Φ∗f〉w =

∫

exp(φx) δ†ma f dχ, x ∈ E. (18)

Proof. The equality (13) yields 〈x | a〉mψ∗n−m(x) =

⟨δ∗ma,nx⊗n | ψn−m

⟩w

forall n ≥ m. By Theorem 2 for any f =

∑n fn ∈ H2

χ there exists a unique


ψ =⊕

n ψn in Γw with ψn ∈ E�n such that Φ∗f = ψ and fn = ψ∗n. Summing

over all m ∈ Z+ and n ≥ m and using (14), we obtain that

Ma∗ f(x) = exp〈x | a〉⟨ε(x) | Φ∗f

⟩w

=∑

m∈Z+

〈x | a〉m

m!

∑

n≥m

ψ∗n−m(x)

=⟨T ∗a ε(x) | Φ∗f

⟩w

=⟨ε(x) | TaΦ

∗f⟩w.

By Theorem 2 and Lemma 3 it follows that the equalities

Mta∗ f(x) = 〈ε(x) | TtaΦ∗f〉w =

∫

exp(φx)T †taf dχ, t ∈ C (19)

hold for all f ∈ H2w. On the other hand, the equalities (14) and (19) yield

dmMta∗ f(x)dtm

∣∣∣t=0

=∫

exp(φx)dmT †

ta

dtm

∣∣∣t=0

f dχ =∫

exp(φx)δ†ma f dχ

for all f ∈ D(δ†ma ). This in turn yields (18). �

7. Commutation Relations

Describe the commutation relations between M†a and T †

b on the Hardy spaceH2

χ.

Theorem 5. For any nonzero a, b ∈ E the commutation relations

M†aT †

b = exp〈a | b〉T †b M†

a , (φaδ†b − δ†

bφa)f = 〈a | b〉f

hold, wherein f belongs to the dense subspace D(φ2b) ∩ D(δ†2

a ) ⊂ H2χ.

Proof. Let us prove that the following equalities hold,

TaMb∗ = exp〈a | b〉Mb∗Ta, (dab∗ − b∗da)f = 〈a | b〉f (20)

where f ∈ D(b∗2) ∩D(d2a). First property follows from the direct calculations:

Mb∗Taf(x) = exp〈x | b〉f(x + a),

TaMb∗ f(x) = f(x + a) exp〈x | b〉 exp〈a | b〉 = exp〈a | b〉Mb∗Taf(x)

for all f ∈ H2w and x ∈ E. For any f ∈ D(b∗2) ∩ D(d2a) and t ∈ C, we have

d2

dt2TtaMtb∗ f

∣∣t=0

=[d2aTtaMtb∗ f + 2daTtab

∗Mtb∗ f + Ttab∗2Mtb∗ f

]t=0

= (d2a + 2dab∗ + b∗2)f .


On the other hand, differentiating again, we have

d

dtTtaMtb∗ f

∣∣t=0

=[ d

dtexp〈ta | tb〉Mtb∗Ttaf + exp〈ta | tb〉 d

dtMtb∗Ttaf

]

t=0,

(d2a + 2dab∗ + b∗2)f =d

dt

[ d

dtTtaMtb∗ f

]

t=0=[ d2

dt2exp〈ta | tb〉Mtb∗Ttaf

+ 2d

dtexp〈ta | tb〉 d

dtMtb∗Ttaf

+ exp〈ta | tb〉 d2

dt2Mtb∗Ttaf

]

t=0

= 2〈a | b〉f + (d2a + 2b∗da + b∗2)f .

This yields (20) where D(b∗2) ∩ D(d2a) contains the dense subspace in H2w of

all polynomials f generating by finite sums Φ∗(f) =⊕

n ψn ∈ Γw.From Mb∗ f(x) = 〈ε(x) | TbΦ

∗f〉w it follows that ITb = Ma∗ I with I :=FΦ. Thus, T †

b = ΦTbΦ∗ = ΦI−1Mb∗ IΦ∗ = F−1Mb∗F . Using that M†

a =F−1TaF with F−1 : H2

w → H2χ and applying (20), we obtain

M†aT †

b = F−1TaMb∗F = exp〈x | b〉F−1Mb∗TaF = exp〈x | b〉T †b M†

a ,

(φaδ†b − δ†

bφa)f = F−1(dab∗ − b∗da)Ff = 〈a | b〉f

for all f ∈ D(φ2b) ∩ D(δ†2

a ). For any f =∑

n fn ∈ H2χ there exists a unique

ψ =⊕

n ψn in Γw with ψn ∈ E�nw such that the equalities Φ∗f = ψ and fn =

ψ∗n hold. Hence, the following embedding D(φ2

b) ∩ D(δ†2a ) ⊂ H2

χ is dense. �

8. Gauss-Weierstrass Semigroups

Next we show that the 1-parameter Gauss-Weierstrass semigroups on theHardy space H2

χ can be well described by shift and multiplicative groups (aclassic case can be found in [7, n.4.3.2]). For this purpose we use the Gaussiankernel

gr(τ) =1√4πr

exp(−τ2

4r

), τ ∈ R, r > 0.

Theorem 6. The 1-parameter Gauss-Weierstrass semigroups{W

δ†a

r : r > 0}

and{W φa

r : r > 0}, defined on the Hardy space H2

χ for any nonzero a ∈ Eas

Wδ†a

r f =∫

R

gr(τ)T †τaf dτ and W φa

r f =∫

R

gr(τ)M†τaf dτ, f ∈ H2

χ, (21)

are generated by δ†2a and φ2

a, respectively.


Proof. First it is sufficient to prove that the axillary 1-parameter families oflinear operators over H2

w

Ga∗r f =

∫

R

gr(τ)Mτa∗ f dτ and G∂ar f =

∫

R

gr(τ)Tτaf dτ, f ∈ H2w (22)

can be generated by a∗2 and d2a and satisfy the semigroup property. Propertiesof Gaussian kernel yield∫

R

gr(τ)τ2k dτ =1

2√

πr

∫

R

e− τ24r τ2kdτ

∣∣∣τ=2

√rυ

=(2

√r)2k

√π

∫

R

e−υ2υ2k dυ

=22krk

√π

Γ(

2k + 12

)

=2(2k − 1)!(k − 1)!

rk, k ∈ N.

We can rewrite Ga∗r f on the dense subspace

{f ∈ H2

w : exp(τa∗)f ∈ H2w

}as

Ga∗r f =

∫

R

gr(τ) exp(τa∗)f dτ =∑

l∈Z+

a∗lf

l!

∫

R

gr(τ)τ l dτ

=∑

k∈Z+

2(2k − 1)!(k − 1)!

rka∗2kf

(2k)!=

∑

k∈Z+

rka∗2kf

k!= exp(ra∗2)f

By first equality in (22) the family Ga∗r can be extended to the convolution

gr � f :=∫

R

gr(τ)Mτa∗ f dτ, f ∈ H2w

(dependent on a) over the whole space H2w. Thus, to show that the semigroup

property holds, it suffices to show that

gr+s � f = Ga∗r+sf = (Ga∗

r ◦ Ga∗s )f = gr � (gs � f) = (gr ∗ gs) � f .

But this straightly follows from the known convolution equality gr+s = gr ∗gs.Further, using the equality T †

a = F−1Ma∗F we obtain that

Wδ†a

r f =∫

R

gr(τ)F−1Mτa∗Ff dτ = F−1Ga∗r Ff

for all f ∈ H2χ. By Theorem 4 it follows that

dWδ†a

r f

dr

∣∣∣r=0

= F−1Ga∗r f

dr

∣∣∣r=0

= F−1a∗2f = δ†2a f

for all f ∈ D(δ†2a ), since f ∈ D(a∗2) and δ†2

a = F−1a∗2F . Hence, the case of

semigroup Wδ†a

r is proven.Similar reasonings can be applied to the semigroup G∂a

r . As a result, weobtain that the equalities W φa

r = F−1G∂ar F and φ2

a = F−1d2aF hold. �


9. Complexified Infinite-Dimensional Heisenberg Group

Let us give yet another application. Consider an infinite-dimensional analogof the Heisenberg group over C. Namely, let us define the group G of uppertriangular matrix-type elements

X(a, b, t) =

⎡

⎣1 a t0 1 b0 0 1

⎤

⎦ , t ∈ C, a, b ∈ E

with unit X(0, 0, 0) and multiplication⎡

⎣1 a t0 1 b0 0 1

⎤

⎦

⎡

⎣1 a′ t′

0 1 b′

0 0 1

⎤

⎦ =

⎡

⎣1 a + a′ t + t′ + 〈a | b′〉0 1 b + b′

0 0 1

⎤

⎦ .

Obviously, X(a, b, t)−1 = X(−a,−b,−t + 〈a | b〉).We will now describe an irreducible linear representation of the group G.

For this purpose we will use the algebra H of quaternions γ = α1 + α2i + β1j+ β2k = (α1 + α2i) + (β1 + β2i)j = α + βj as pairs of complex numbers(α, β) ∈ C

2 with α = α1 + α2i, β = β1 + β2i ∈ C and αı, βı ∈ R (ı = 1, 2)where basis elements in R

4 satisfy the relations i2 = j2 = k2 = ijk = −1,k = ij = −ji, ki = ik = j. Thus, H = C ⊕ Cj is a vector space over C [25].Denote β := �γ where γ = α + βj.

Let EH = E ⊕ Ej be the Hilbert space with H-valued scalar product

〈p | p′〉 = 〈a + bj | a′ + b′j〉 = 〈a | a′〉 + 〈b | b′〉 + [〈a′ | b〉 − 〈a | b′〉] jwhere p = a + bj with a, b ∈ E (similarly, for p′ = a′ + b′j). Hence,

�〈p | p′〉 = 〈a′ | b〉 − 〈a | b′〉, �〈p | p〉 = 0.

The following theorem describes a representation of the above infinite-dimensional Heisenberg group which can be seen as an analog of the Weyl–Schrodinger representation

Theorem 7. The linear representation of G over H2χ

W † : G X(a, b, t) −→ exp[t +

12〈a | b〉

]T †a M†

b

is well defined and irreducible.

Proof. First we prove that the following operator representation

W : G X(a, b, t) −→ exp[t +

12〈a | b〉

]Ma∗Tb

into the algebra of all bounded linear operator on H2w is well defined and

irreducible. Consider the auxiliary group C × EH with the multiplication

(t, p)(t′, p′) =(t + t′ − 1

2�〈p | p′〉, p + p′

)


for all p = a + bj, p′ = a′ + b′j ∈ EH. It is related to G via the mapping

G : X(a, b, t) −→(t − 1

2〈a | b〉, a + bj

).

Check that G is a group isomorphism. In fact,

G (X(a, b, t)X(a′, b′, t′)) = G (X(a + a′, b + b′, t + t′ + 〈a | b′〉))

=(t + t′ + 〈a | b′〉 − 1

2[〈a + a′ | b + b′〉] , (a + a′) + (b + b′)j

)

=(t + t′ − 1

2[〈a | b〉 + 〈a′ | b′〉

]+

12[〈a | b′〉 − 〈a′ | b〉

], (a + a) + (b + b′)j

)

=(t − 1

2〈a | b〉, a + bj

)(t′ − 1

2〈a′ | b′〉, a′ + b′j

)

= G (X(a, b, t))G (X(a′, b′, t′)) .

Now let us check that the Weyl-like operator

W (p) = exp[12〈a | b〉

]Ma∗Tb, p = a + bj

on the space H2w satisfies the commutation relation

W (p + p′) = exp[

− 12�〈p | p′〉

]W (p)W (p′).

In fact, using (20), we obtain

exp[12〈a | b′〉 − 1

2〈a′ | b〉

]W (p)W (p′)

= exp[12〈a | b〉 +

12〈a′ | b′〉

]exp

[12〈a | b′〉 − 1

2〈a′ | b〉

]Ma∗TbMa′∗Tb′

= exp[12〈a + a′ | b + b′〉

]Ma∗+a′∗Tb+b′ = W (p + p′).

As a consequence, the mapping I : C × EH (t, p) −→ exp(t)W (p) is a groupisomorphism. So, W is also a group isomorphism as a composition of the groupisomorphisms I and G .

Let us check irreducibility. If there exists an element x0 =/ 0 in E and aninteger n > 0 such that

exp[t +

12〈a | b〉

]e〈a|x〉 [x∗

0(x + b)]n = 0 for all x, a, b ∈ E

then x0 = 0. This gives a contradiction. Hence the representation W is irre-ducible. Finally, using that

exp[t +

12〈a | b〉

]T †a M†

b = F−1(

exp[t +

12〈a | b〉

]Ma∗Tb

)F ,

we conclude that the group representation W † = F−1W F is irreducible. �


Acknowledgements

I am grateful to Referee for valuable suggestions which improved this article.

Open Access. This article is distributed under the terms of the Creative Com-mons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changeswere made.

References

[1] Beltita, I., Beltita, D.: On Weyl calculus in infinitely many variables. Geom.Methods Phys. In: Kielanowski, P., Buchstaber, V., Odzijewicz, A., Schlichen-maier, M., Voronov, T. (eds.) XXIX Workshop 2010, pp. 95–104 (2010)

[2] Beltita, I., Beltita, D.: Representation of nilpotent Lie groups via measurabledynamical systems. Geom. Methods Phys. In: XXXIV Workshop 2015 Trendsin Mathematics, pp. 95–104 (2015)

[3] Beltita, I., Beltita, D., Mantoiu, M.: On Wigner transforms in infinite dimen-sions. J. Math. Phys. 57(021705), 1–13 (2016)

[4] Berezanski, Y.M., Kondratiev, Y.G.: Spectral Methods in Infinite-DimensionalAnalysis. Springer, Berlin (1995)

[5] Borodin, A., Olshanski, G.: Harmonic analysis on the infinite-dimensional uni-tary group and determinantal point processes. Ann. Math. 161(3), 1319–1422(2005)

[6] Bourbaki, N.: Integration II. Springer, Berlin (2004)

[7] Butzer, P.L., Berens, H.: Semi-groups of Operators and Approximation.Springer, Berlin (1967)

[8] Carando, D., Dimand, V., Muro, S.: Coherent sequences of polynomial ideals onBanach spaces. Math. Nachr. 282(8), 1111–1133 (2009)

[9] Cole, B., Gamelin, T.W.: Representing measures and Hardy spaces for the infi-nite polydisk algebra. Proc. Lond. Math. Soc. 53, 112–142 (1986)

[10] Dwyer, A.W.: Partial differential equationa in Fischer-Fock spaces for theHilbert–Schmidt holomorphy type. Bull. Am. Soc. 77, 725–730 (1971)

[11] Floret, K.: Natural norms on symmetric tensor products of normed spaces. NoteMat. 17, 153–188 (1997)

[12] Holmes, I., Sengupta, A.N.: The Gaussian Radon transform in the classicalWiener space. Commun. Stoch. Anal. 8(2), 247–268 (2014)

[13] Klenke, A.: Probability Theory. A Comprehensive Course. Springer, Berlin(2008)

[14] Lopushansky, O., Zagorodnyuk, A.: Hilbert spaces of analytic functions of infin-itely many variables. Ann. Pol. Math. 81(2), 111–122 (2003)

[15] Lopushansky, O.: The Hilbert–Schmidt analyticity associated with infinite-dimensional unitary groups. Results Math. 71(1), 111–126 (2017)

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[16] Neeb, K.-H.: Holomorphy and Convexity in Lie Theory, vol. 28. De GruyterExpositions in Mathematics, Berlin (2000)

[17] Neeb, K.-H., Ørted, B.: Hardy spaces in an infinite dimensional sett. In: Doebner,H.D. (ed.) Lie Theory and Its Applications in Physics, pp. 3–27. World ScientificPublishing, Singapore (1998)

[18] Neretin, Y.A.: Hua-type integrals over unitary groups and over projective limitsof unitary groups. Duke Math. J. 114(2), 239–266 (2002)

[19] Okada, S., Okazaki, Y.: Projective limit of infinite radon measures. J. Aust.Math. Soc. 25(A), 328–331 (1978)

[20] Olshanski, G.: The problem of harmonic analysis on the infinite-dimensionalunitary group. J. Funct. Anal. 205(2), 464–524 (2003)

[21] Petersson, P.: Hypercyclic convolution operators on entire functions of Hilbert–Schmidt holomorphy type. Ann. Math. Blaise Pascal 8(2), 107–114 (2001)

[22] Pinasco, D., Zalduendo, I.: Integral representations of holomorphic functions onBanach spaces. J. Math. Anal. Appl. 308, 159–174 (2005)

[23] Stroock, D.W.: Abstract Wiener space, revisited. Commun. Stoch. Anal. 2(1),145–151 (2008)

[24] Tomas, E.: On Prohorov’s criterion for projective limits. Oper. Theory Adv.Appl. 168, 251–261 (2006)

[25] Ward, J.: Quaternions and Cayley Numbers: Algebra and Applications. KluwerAcademic Publishers, Dordrecht (1997)

[26] Yamasaki, Y.: Projective limit of Haar measures on O(n). Publ. Res. Inst. Math.Sci. 8, 141–149 (1972)

Oleh LopushanskyFaculty of Mathematics and Natural SciencesUniversity of Rzeszow1 Pigonia Str.35-310 RzeszowPolande-mail: [email protected];

[email protected]

Received: April 2, 2017.

Accepted: September 7, 2017.

paley-wiener isomorphism over infinite-dimensional unitary...

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