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-Schr\"odinger irreducible representations of complexified infinite-dimensional Heisenberg groups.


Introduction
The work deals with the Hardy space H 2 χ of square-integrable complex-valued functions with respect to a probability measure χ over the infinite-dimensional unitary group U (∞) := Ť {U (m) : m ∈ N}, extended by unit ½, which irreducibly acts on a separable complex Hilbert space E. Here, U (m) is the subgroup of unitary (m × m)-matrices endowed with Haar's measure χ m . 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, Neretin 2002], [20, Olshanski 2003]). The measure χ is invariant under right actions [20, n.4]. We refer to [23, Yamasaki 1974], [5, Borodin and Olshanski 2005] 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, Cole and Gamelin 1986], [17, Ørsted and Neeb 1998]. Now, we briefly describe results. Using a unitarily weighted symmetric Fock space Γ w , defined by E and χ, we find an orthogonal basis in H 2 χ of Hilbert-Schmidt polynomials such that the conjugate-linear mapping Φ : Γ w → H 2 is a surjective isometry. This allows us to establish in Theorem 4.2 an integral formula for a Fock-symmetric F-transform F : 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 from E over the Hardy space H 2 χ by shift and multiplicative groups. Theorem 6.1 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 6.2 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 7.1 suitable commutation relations are stated.
Applications to the Gauss-Weierstrass-type semigroups on H 2 χ are shown in Theorem 8.1. An another application to linear representations of complexified infinite-dimensional Heisenberg groups on H 2 χ in a Weyl-Schrödinger form is given in Theorem 9.1. Infinite-dimensional Heisenberg groups was considered in [16, Neeb 2000] by using reproducing kernel Hilbert spaces. The Schrödinger representation of infinite-dimensional Heisenberg groups on L 2 γ with respect to a Gaussian measure γ over a real Hilbert space is described in [3, I.Beltiţȃ, D.Beltiţȃ and M.Mȃntoiu 2016] (see also earlier publications [1,2]).

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 : (1), . . . , σ(n)} runs through all permutations. The symmetric algebraic Fock space is defined as the orthogonal sum 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 Γ h = 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 · w = · | · 1/2 w where scalar product · | · w is determined via the orthogonal relations Denote by E ⊙n w and Γ w the appropriate completions of E ⊙n alg and Γ alg , respectively. For any ı ∈ N l(λ) * there corresponds in E ⊙n w the d-dimension subspace with d = C −1 |λ|,l(λ) , spanned by elements e ⊙λ ı : λ ∈ Y n . The Hilbertian orthogonal sum Γ w = à n∈Z + E ⊙n w endowed with · | · w we will call unitarily weighted symmetric Fock space. Let x = ř e k x k be the Fourier series of x ∈ E with coefficients x k = x | e k . We assign to 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 (2.1) and (2.4) Similarly, for the subspace H 2,n w which is uniquely determined by E ⊙n w , since {x ⊗n : x ∈ E} is total in E ⊙n w . 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, Sec. 1.5]) Thus, the conjugate-linear 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 = À ψ n ∈ Γ w , (ψ n ∈ E ⊙n w ) has the Taylor expansion at zero The function ψ * is entire Hilbert-Schmidt analytic [15, n.5], [14, n.2]. Note that analytic functions of Hilbert-Schmidt types were considered in [10] [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 embedding U 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 Following [18, and therefore the projective limit lim ← − χ m exists on U π via the well known Prohorov theorem [6, Thm IX.52]. Moreover, it is a Radon probability measure concentrated on U π [25, Thm 4.1]. By the known portmanteau theorem [13,Thm 13.16] and the Fubini theorem, the invariance of Haar measures χ m together with (3.2) yield the 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 The embedding L ∞ χ L 2 χ holds, moreover, f χ ≤ f ∞ for all f ∈ L ∞ χ . To given the E-valued mapping U π ∋ u → π −1 (u)e 1 , we can well-define the Borel χ-essentially bounded functions in the variable u ∈ U π , i.e., φ x ∈ L ∞ χ for any φ x (u) = π −1 (u)e 1 | x with x ∈ E. By formula (2.5) to every e ⊙λ ı ∈ e ⊙Yn there uniquely corresponds the Borel function from L ∞ in the variable u ∈ U π . It follows that the orthogonal basis e ⊙Y of elements e ⊙λ 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,Thm 3.2].
Theorem 3.1. The system of Borel functions φ Y forms an orthogonal basis in H 2 χ such that 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 3.1 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 isometric conjugate-linear mapping Φ : Γ w → H 2 χ . The adjoint mapping Φ * : for any ψ ∈ Γ w . In particular, the equality Φx = ř 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 φ x can be seen as a group analog of the Paley-Wiener map (see e.g. [12, n.4.4] or [24]).

It directly follows that
x ∈ E and its Taylor coefficients at origin have the integral representations Proof. First recall that the Γ w -valued function ε(·) is entire analytic on E, therefore p 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 , Applying Φ * to f in this decomposition and substitutingf λ,ı,n into p f , we obtain where the last equality is valid by Lemma 4.1. It particularly follows that for y = αx, Differentiating p 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, enough 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 h , λ ∈ Y n which follows from Theorem 3.1, taking into account the inequality (2.1). Using subsequently that E ⊙n h is reflexive, we obtain that its adjoint operator j * n : E ⊙n h → E ⊙n w is a contractive injection with dense range. Thus, the mapping j n is also injective. Moreover, E ⊙n with respect to e ⊙λ ı ⊗ e ⊙µ  for all λ, µ ∈ Y, ı ∈ N l(λ) ,  ∈ N l(µ) such that |λ| = m, |µ| = n − m. Using (5.1), we have As above, it implies that the mapping j m ⊗ j n−m : 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 (5.2) follows from the binomial formula for symmetric tensor elements (x + ta) ⊗n = ř n m=0 n m (ta) ⊗m ⊙ x ⊗(n−m) . Put δ 0 a,n = 1. If a = 0 then δ m 0,n = 0. Summing over n ≥ m with coefficients 1/(n − m)!, we get This series is convergent, since by Lemma 5.1 and (2.4) the inequality holds. From (5.3) and the tensor binomial formula mentioned above it follows that Summing over n ∈ Z + with coefficients 1/n! and using (5.3), we obtain The inequalities (2.4) and (5.4) yield T a ε(x) 2 w ≤ exp a 2 ε(x) 2 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.
We define the adjoint operators δ * m a,n : Using δ * m a,n , we can define the exponential annihilation group by the equalities for all a, x ∈ E. Taking into account Lemma 5.2, we obtain the following claim.
Lemma 5.3. The exponential annihilation group T * a defined by (5.6) possesses a unique linear extension T * a : for all a, b ∈ E.

Intertwining properties of F-transform
Let us define on the space H 2 It can be considered as a linear representation of the additive group from E. By Lemma 4.1 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 densely-defined. 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.
The additive group contained in E may be also linearly represented on H 2 w as the shift group T a p f (x) = p f (x + a), f ∈ H 2 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, Theorem 6.2. For every f ∈ H 2 χ the following equality holds, M a * F(f ) = F(T † a f ), a ∈ E, that is, the F-transform is an intertwining operator for the groups M a * on H 2 w and T † a on H 2 χ . Moreover, for every f ∈ D(δ †m a ) = f ∈ H 2 χ : δ †m a f ∈ H 2 χ (m ∈ N) and a nonzero a ∈ E,

+2
(dependent on a) over the whole space H 2 w . Thus, to show that the semigroup property holds, it suffices to show that But this straightly follows from the known convolution equality g r+s = g r * g s . Further, using the equality T † a = F −1 M a * F we obtain that Ff for all f ∈ H 2 χ . By Theorem 6.2 it follows that for all f ∈ D(δ †2 a ), since p f ∈ D(a * 2 ) and δ †2 a = F −1 a * 2 F. Hence, the case of semigroup W δ † a r is proven.
Similar reasonings can be applied to the semigroup G ∂a r . As a result, we obtain that the equalities Wφ a r = F −1 G ∂a r F andφ 2 a = F −1 d 2 a F hold.
Proof. First we prove that the following operator representation