The Hilbert–Schmidt Analyticity Associated with Infinite-Dimensional Unitary Groups

The article is devoted to the problem of Hilbert–Schmidt type analytic extensions in Hardy spaces over the infinite-dimensional unitary group endowed with an invariant probability measure. Reproducing kernels of Hardy spaces, integral formulas of analytic extensions and their boundary values are considered.


Introduction
The paper deals with the problem of Hilbert-Schmidt type analytic extensions in the Hardy space H 2 χ of complex functions over the infinite-dimensional group U (∞) = {U (m): m ∈ N} endowed with an invariant probability measure χ where U (m) are subgroups of unitary m×m-matrices. The measure χ is defined as a projective limit χ = lim ← − χ m of the Haar probability measures χ m on U (m). Moreover, χ is supported by a projective limit U = lim ← − U (m) and is invariant under the right action of U 2 (∞) := U (∞) × U (∞) on U.
A goal of this work is to find integral formulas for Hilbert-Schmidt analytic extensions of functions from H 2 χ and to describe their radial boundary values on the open unit ball in a Hilbert space E where U (∞) acts irreducibly.
The measure χ on U was described by Olshanski [13] and Neretin [12]. The notion U is related to Pickrell's space of a virtual Grassmannian [16]. Hardy spaces in infinite-dimensional settings were discussed in the works of Cole and Gamelin [5], Ørsted and Neeb [14]. Spaces of analytic functions of Hilbert-Schmidt holomorphy types were considered by Dwyer III [6] and Petersson [15].

Background on Invariant Measure
Let U (m) (m ∈ N) be the group of unitary (m × m)-matrices. We endow U (∞) = U (m) with the inductive topology under every continuous inclusion U (m) U (∞) which assigns to any u m ∈ U (m) the matrix The right action over U (∞) is defined via Following [12,13], every u m ∈ U (m) with m > 1 can be written as It was proven that the Livšic-type mapping (which is not a group homomorphism) is Borel and surjective. Consider the projective limit U = lim ← − U (m) taken with respect to π m m−1 . The embedding ρ : U (∞) U assigns to every u m ∈ U (m) the stabilized sequence u = (u k ) k∈N (see [13, n.4]) so that Vol. 71 (2017) The Hilbert-Schmidt Analyticity 113 where the projections π m : U u −→ u m ∈ U (m) such that π m m−1 • π m = π m−1 are surjective and π m k := π k+1 k • · · · • π m m−1 for k < m. Using (2.1), the right action of U 2 (∞) over U can be defined as where m is so large that g = (v, w) ∈ U 2 (m) (see [13,Def 4.5]). We endow every group U (m) with the probability Haar measure χ m . It is known [12,Thm 1.6]  Following [13,Lem. 4.8], [12, n.3.1], via of the Kolmogorov consistency theorem we uniquely define on U the probability measure χ which is the projective limit under the mapping (2.2), i.e., we put is the projective limit with respect to π m m−1 | U (m) then U\U is χ-negligible, because χ m is zero on U (m) \ U (m) for any m.
A complex-valued function on U is called cylindrical if it has the form f = f m • π m for a certain m ∈ N and a complex function f m on U (m) [13,Def. 4.5]. By L ∞ χ we denote the closed linear hull of all cylindrical χ-essentially bounded Borel functions endowed with the norm f L ∞ χ = ess sup u∈U |f (u)|. The measure (2.5) is a probability measure and is U 2 (∞)-invariant under the right actions (2.4) over U [12,Prop. 3.2]. Moreover, this measure is Radon so that and it satisfies the property: Using the invariance property (2.6) and the Fubini theorem (see [11,Lem. 2 The closed linear hull of cylindrical complex functions endowed with the norm f L 2

Hardy Spaces
Throughout the paper E is a separable complex Hilbert space with an orthonormal basis {e k : k ∈ N}, scalar product · | · and norm · = · | · 1/2 . So, for any element x ∈ E the following Fourier decomposition holds, Let E ⊗n be the complete nth tensor power of E endowed with the scalar product and norm (1), . . . , σ(n)} runs through all n-elements permutations, the symmetric complete nth tensor power E n is defined to be a codomain of the orthogonal projector Note that . . , λ m ) ∈ N m be a partition of an integer n ∈ N with m ≤ n and λ 1 ≥ λ 2 ≥ · · · λ m > 0, i.e., |λ| = n where |λ| := λ 1 + · · · + λ m . We identify partitions with Young diagrams. By (λ) = m we denote the length of λ defined as the number of rows in λ. Let Y denote all Young diagrams and Y n := {λ ∈ Y : |λ| = n}. Assume that Y includes the empty partition ∅ = (0, 0, . . .).
An orthogonal basis in E n is formed by the system of symmetric tensor products (see e.g.
In what follows, we will use the fact that for every ψ ∈ E n one can uniquely define the so-called Hilbert-Schmidt n-homogenous polynomial Vol. 71 (2017) The Hilbert-Schmidt Analyticity 115 In fact, the polarization formula for symmetric tensor products (see [8, 1.5]) . . . , z n ∈ E) implies that the n-homogenous polynomial x ⊗n | ψ is uniquely determines ψ, because the set of all z 1 · · · z n is total in E n . Using the embedding ( where ε k := e * k • ζ. Using ζ, we may define the E n -valued Borel mapping The following assertion, which is a consequence of the polarization formula · · · e ⊗λm ım , indexed by λ = (λ 1 , . . . , λ m ) ∈ Y and ı = (ı 1 , . . . , ı m ) ∈ N m * with m = (λ), uniquely defines the appropriate system of χ-essentially bounded cylindrical functions in the variable u ∈ U that possess continuous restrictions to U .

Theorem 3.2.
For any ı ∈ N m * and ψ, φ ∈ E n ı , the following equality holds, As a consequence, given for all v ∈ U (ı) and u ∈ U. Using (2.7) with U (ı) instead of U (m), we have for all ψ, φ ∈ E n ı . It is clear that Next we show that A commutes with all operators w ⊗n ∈ L (E n ı ) with w ∈ U (ı) acting as w ⊗n x ⊗n = (wx) ⊗n , (x ∈ E ı ). Invariance properties (2.6) of χ ı under the right action (2.4) yield where w −1 ∈ U (ı) is the hermitian adjoint matrix of w. Hence, the equality Vol. 71 (2017) The Hilbert-Schmidt Analyticity 117 holds. Let us check that the operator A, satisfying the condition (3.8), is proportional to the identity operator on E ⊗n ı . To this end we form the nth tensor power of the unitary group U (ı), Clearly, [U (ı)] ⊗n is a unitary group over E n ı . Let us check that the corresponding unitary representation is irreducible. This means that there is no subspace in E n ı other than {0} and the whole space which is invariant under the action of [U (ı)] ⊗n .
By Lemma 3.1 the elements wρ −1 (u) act transitively on S(∞). Hence, by nhomogeneity, we obtain x ⊗n | ψ = 0 for all x ∈ E ı . Applying the polarization formula (3.3), we get ψ = 0. Hence, (3.9) is irreducible. Thus, we can apply to (3.9) the Schur lemma [10, Thm 21.30]: a nonzero matrix which commutes with all matrices of an irreducible representation is a constant multiple of the unit matrix. As a result, we obtain that the operator A, satisfying (3.8), is proportional to the identity operator on E n ı i.e. A = α (n,ı) 1 E n ı with a constant α (n,ı) > 0. It follows that In particular, the subsystem of cylindrical functions ε λ ı with a fixed ı ∈ N m * is orthogonal in L 2 χ , because the corresponding system of tensor products e λ ı indexed by λ ∈ Y n with (λ) = m forms an orthogonal basis in E n ı . It remains to note that the set of all indices ı = (ı 1 , . . . , ı m ) ∈ N m * with all m = (λ) is directed with respect to the set-theoretic embedding, i.e., for any ı, ı there exists ı so that ı ∪ ı ⊂ ı . This fact and the above reasoning imply that the whole system ε Y is also orthogonal in L 2 χ . Taking into account (3.2), we can choose φ n = ψ n = ε λ ı n!/λ! in (3.10). As a result, we obtain The well known formula [18, 1.4.9] for the unitary m-dimensional group gives Using the last two formulas, we arrive at the relation Combining (3.7) and (3.11), we get (3.5) and, as a consequence, (3.6).

Definition 3.3.
By H 2 χ we denote the Hardy space over U (∞) defined as the L 2 χ -closure of the complex linear span of the orthogonal system ε Y .

Reproducing Kernels
Let us construct the reproducing kernel of H 2 χ . We refer to [19] for the basic definitions and properties of reproducing kernels.