Paley-Wiener Isomorphism Over Infinite-Dimensional Unitary Groups

An analog of the Paley-Wiener isomorphism for the Hardy space with an invariant measure over infinite-dimensional unitary groups is described. This allows us to investigate on such space the shift and multiplicative groups, as well as, their generators and intertwining operators. We show applications to the Gauss-Weierstrass semigroups and to the Weyl–Schrödinger irreducible representations of complexified infinite-dimensional Heisenberg groups.


Introduction
The work deals with the Hardy space H 2 χ of square-integrable C-valued functions with respect to a probability measure χ over the infinite- In what follows, U (∞) is densely embedded via a universal mapping π into the space of virtual unitary matrices U = lim ← − U (m) defined as the projective limit under Livšic's mappings π m+1 m : U (m + 1) → U (m). The projective limit χ = lim ← − χ m , such that each image-measure π m+1 m (χ m+1 ) is equal to χ m , 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]. We refer to [5,26] for applications of χ to stochastic processes. Needed properties of Hardy spaces H 2 χ can be found in [15]. Various cases of Hardy spaces in infinite-dimensional settings were considered in [9,17]. Now, we briefly describe results. Using a unitarily weighted symmetric Fock space (Γ w , · | · w ) with a canonical orthogonal basis of symmetric tensor products {e λ ı } of basis elements {e m } ⊂ E indexed by Young diagrams λ and normalized by measure χ, we find an orthogonal basis of polynomial {φ λ ı } in H 2 χ such that the conjugate-linear mapping Φ : Γ w → H 2 χ is a surjective isometry with one-to-one correspondence e λ ı φ λ ı . This allows us to establish in Theorem 2 an integral formula for a Fock-symmetric Ftransform where the Hilbert space H 2 w , uniquely determined by Γ w , consists of Hilbert-Schmidt analytic entire functions on E. Thus, the F-transform acts as an analog of the Paley-Wiener isomorphism over infinite-dimensional groups. Furthermore, we investigate two different representations of the additive group (E, +) over the Hardy space H 2 χ by shift and multiplicative groups. Theorem 3 states that the F-transform is an intertwining operator between the multiplication group M † a on H 2 χ and the shift group T a on H 2 w . On the other hand, Theorem 4 shows that F is the same between the shift group T † a on H 2 χ and the multiplication group M a * on H 2 w . Integral formulas describing interrelations between their generators are established. In Theorem 5 suitable commutation relations are stated.
Applications to the Gauss-Weierstrass-type semigroups on H 2 χ are shown in Theorem 6. Another application to linear representations of complexified infinite-dimensional Heisenberg groups on H 2 χ in a Weyl-Schrödinger form is given in Theorem 7.
Infinite-dimensional Heisenberg groups was considered in [16] by using reproducing kernel Hilbert spaces. The Schrödinger representation of infinitedimensional Heisenberg groups on L 2 γ with respect to a Gaussian measure γ over 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 following simple relations in the Hardy space In result, T a Mb = exp(ab) MbT a for all a, b ∈ C and the Weyl-Schrödinger representation of Heisenberg's group from Theorem 7 retains a classic form.
The case H 2 χ over m-dimensional group U (m) is similar with a proviso that the weighted Fock space Γ w is normalized by e λ ı w = n+m−1 n −1/2 where n = |λ| is a homogeneity degree of the basis polynomial φ λ ı in H 2 χ . Note that the normalization e λ ı w = n! −1/2 with n = |λ| leads to the case of Segal-Bargmann's space H 2 γ with standard centered probability Gaussian measure γ on C m .

Hilbert-Schmidt Analyticity
Let E stand for a separable complex Hilbert space with scalar product · | · norm · and a fixed orthonormal basis {e k : k ∈ N}. Denote by E ⊗n alg = E⊗ n times · · · ⊗E (n ∈ N) its algebraic tensor power consisted of the linear span of elements ψ n = x 1 ⊗ · · · ⊗ x n with x i ∈ E (i = 1, . . . , n). Set x ⊗n := x⊗ n times · · · ⊗x. The symmetric algebraic tensor power E n alg = E · · · E is defined to be the range of the projector s n : E ⊗n (1), . . . , σ(n)} runs through all permutations. The symmetric algebraic Fock space is defined as the algebraic direct sum Γ alg = n∈Z+ E n alg with E 0 alg = C. Let E ⊗n h := E ⊗ h · · · ⊗ h E be the completion of E ⊗n alg by Hilbertian norm ψ n h = ψ n | ψ n 1/2 h with ψ n | ψ n h = x 1 | x 1 · · · x n | x n . Denote by E n h the range of continuous extension of s n on E ⊗n h . As usual, the symmetric Fock space is defined to be the Hilbertian direct sum Γ h = n∈Z+ E n h . Denote by λ = (λ 1 , . . . , λ m ) ∈ Z m + with λ 1 ≥ λ 2 ≥ · · · ≥ λ m a partition of n ∈ N, that is, n = |λ| where |λ| := λ 1 + · · · + λ m . Any λ may be identified with Young's diagram of length l(λ) = m. Let Y denote all diagrams and The spaces E n alg and Γ alg may be generated by the basis of symmetric tensors Let us define a new Hilbertian norm on Γ alg by the equality Denote by E n w and Γ w the appropriate completions of E n alg and Γ alg , respectively. For any ı ∈ N , spanned by elements e λ ı : λ ∈ Y n . The Hilbertian orthogonal sum the n-homogenous Hilbert-Schmidt polynomial defined via the Fourier coefficients Using the tensor multinomial theorem, we define in Γ w the Fourier decomposition of exponential vectors (or coherent state vectors) with respect to the basis e Y . It is convergent in Γ w in view of (1) and Particulary, (4) implies that the function E x → ε(x) ∈ Γ w is entire analytic.
Every function ψ * is entire analytic as the composition of ε(·) with · | ψ w . The subspace in H 2 w of n-homogenous Hilbert-Schmidt polynomials is defined to be Evidently, The last totality follows from the polarization formula for symmetric tensor products which is valid for all e λ ı ∈ e Yn (see e.g. [11, Sect. 1.5]) Thus, the conjugatelinear isometries ψ → ψ * from Γ w onto H 2 w and from E n w onto H 2,n w hold. In conclusion, we can notice that every analytic function ψ * ∈ H 2 w determined by ψ = ψ n ∈ Γ w , (ψ n ∈ E n w ) has the Taylor expansion at zero 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 considered in [10,14,21]. More general classes of analytic functions associated with coherent sequences of polynomial ideals were described in [8].

Hardy Space Over U (∞)
In what follows, we endow each group U (m) with the probability Haar measure χ m and assume that U (m) is identified with its range with respect to The projective limit U := lim ← − U (m) under π m+1 m has surjective Borel projections π m : U u → u m ∈ U (m) such that π m = π m+1 m • π m+1 . Consider a universal dense embedding π : U (∞) U which to every u m ∈ U (m) assigns the stabilized sequence u = (u k ) such that (see [20, n.4]) where π m k := π k+1 k • . . . • π m m−1 for k < m and π m k is identity mapping for k ≥ m. On its range π(U (∞)), endowed with the Borel structure from U, we consider the inverse mapping . Hence, χ satisfies the condition and therefore the projective limit lim ← − χ m exists on U π via the well known Prohorov theorem [6, Theorem IX.52]. Moreover, it is a Radon probability measure concentrated on U π [24, Theorem 4.1]. By the known portmanteau theorem [13,Theorem 13.16] and Fubini's theorem the invariance of Haar measures χ m together with (7) yield the following invariance properties under the right action where L ∞ χ stands for the space of all χ-essentially bounded complex-valued functions defined on U π and endowed with norm f ∞ = ess sup u∈Uπ |f (u)|.
Let L 2 χ be the space of square-integrable C-valued functions f on U π with norm Vol. 72 (2017)

Definition 2.
The Hardy space H 2 χ is defined as the closed complex linear span of φ Y endowed with L 2 χ -norm.
The following assertion is proved in [15, Theorem 3.2].

Theorem 1. The system of Borel functions φ Y forms an orthogonal basis in
Define the subspace H 2,n χ ⊂ H 2 χ for any n ∈ N to be the closed linear span of the subsystem φ Yn . Theorem 1 implies that H 2,n χ ⊥ H 2,m χ in L 2 χ for any n = / m. This provides the orthogonal decomposition

Fock-Symmetric F -Transform
The one-to-one correspondence e λ ı φ λ ı allows us to define via the change of orthonormal bases The suitable Fourier decomposition has the form for any ψ ∈ Γ w . In particular, the equality Φx = x k φ k is valid for all x ∈ E. This gives the equalities Using this, we will examine the composition of Φ with the Γ w -valued function ε : E x → ε(x). Its correctness justifies the following assertion that substantially uses the L ∞ χ -valued function 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]).

takes values in L ∞
χ for all x ∈ E. Proof. Applying Φ to the Fourier decomposition (3), we obtain Proof. First recall that the Γ w -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 φ Y ,

and its Taylor coefficients at origin have the integral representations
Applying Φ * to f in this decomposition and substitutingf λ,ı,n into f , we obtain where the last equality is valid by Lemma 1. It particularly follows that for y = αx, Differentiating f at y = 0 and using the n-homogeneity of derivatives, we obtain Finally, we notice that the isometry H 2 χ H 2 w holds, since the isometry Φ * is surjective. In the case of polynomials we similarly get H 2,n χ H 2,n w .
Note that a different integral formula for analytic functions employing Wiener measures on infinite-dimensional Banach spaces was presented in [22].

Exponential Creation and Annihilation Groups
Let us define the linear mapping j n : E n w → E n h to be the continuous extension of identity mapping acting on the dense subspace E n alg ⊂ E n w ∩ E n h . Such continuous extension j n is a contractive injection with dense range. In fact, it suffices to expand elements from E n w and E n h into the Fourier series with respect to orthogonal basis e Yn and apply the inequality which follows from Theorem 1, taking into account the inequality (1). Using subsequently that E n h is reflexive, we obtain that its adjoint operator As above, it implies that the mapping where injections are contractive and have dense ranges.
has a unique linear extension T a : Γ w ψ → T a ψ ∈ Γ w such that Proof. Let us define the creation operators δ m a,n : E for all a, x ∈ E. Note that the second equality in (10) follows from the binomial formula for symmetric tensor elements (x + ta) ⊗n = Vol. 72 (2017)

Paley-Wiener Isomorphism 2111
This series is convergent, since by Lemma 2 and (4) the inequality holds. From (11) and the tensor binomial formula mentioned above it follows that Summing over n ∈ Z + with coefficients 1/n! and using (11), we obtain The inequalities (4) and (12) w . Taking into account the totality of {ε(x): x ∈ E}, this inequality implies the required inequality on Γ w . It also follows that T a+b = T a T b = T b T a , since δ a+b = δ a + δ b for all a, b ∈ E by linearity of creation operators. This ends the proof.
Using δ * m a,n , we can uniquely define a Γ w -valued function T * a by the equalities for all a, x ∈ E. Taking into account Lemma 3, we obtain the following claim.

Lemma 4.
The Γ w -valued function T * a , defined by (14), possesses a unique linear extension T * a : Γ w ψ → T * a ψ ∈ Γ w such that Definition 3. We will call the Γ w -valued functions T a and T * a in variable a ∈ E the exponential creation and annihilation groups, respectively.

Intertwining Properties of F -Transform
Let us define on the space H 2 χ the multiplicative group M † a : E a → M † a to be M † a f (u) = exp[φ a (u)]f (u), f ∈ H 2 χ , u ∈ U π . It can be considered as a linear representation of the additive group (E, +). By Lemma 1 the function u → exp[φ a (u)] with a fixed a belongs to L ∞ χ . Hence, M † a is continuous on H 2 χ . The generator of the 1-parameter group C t → M † ta coincides with the operator of multiplication by the L ∞ χ -valued function φ a : U π u →φ a (u) where dM † ta /dt| t=0 =φ a . The continuity of E a → exp(φ a ) implies that this 1-parameter group M † ta is strongly continuous on H 2 χ . As a consequnce, its generator (φ a f )(u) = φ a (u)f (u) with domain D(φ a ) = f ∈ H 2 χ :φ a f ∈ H 2 χ is closed and denselydefined. As well, its powerφ m a defined on D(φ m a ) = f ∈ H 2 χ :φ m a f ∈ H 2 χ for any m ∈ N is the same (see, e.g. [7] for details).
The additive group (E, +) may be also linearly represented on H 2 w as the shift group χ , x,a ∈ E. The directional derivative on the space H 2 w along a nonzero a ∈ E coincides with the generator of the 1-parameter shift subgroup C t → T ta , that is, Note that the 1-parameter shift group T ta , which is intertwined with M † ta by the F-transform for all f ∈ D(φ m a ) and x ∈ E. On the other hand, by Theorem 2 we have 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