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 matrix group endowed with an invariant probability measure. An orthogonal basis of Hilbert-Schmidt polynomials, reproducing kernels, integral formulas and boundary values are investigated.


Introduction
The problem of Hilbert-Schmidt type analytic extensions in Hardy spaces H 2 χ of complex functions over the infinite-dimensional unitary matrix group U (∞) = Ť {U (m) : m ∈ N} is investigated. The group U (∞) irreducibility operates on a separable complex Hilbert space E and it endows with a projective limit χ = lim ← − χ m of Haar measures χ m on the m-dimensional unitary groups U (m) so that χ is Radon probability. Furthermore, the measure χ is supported by an appropriate projective limit U = lim ← − U (m) and it is invariant under a right action of U 2 (∞) := U (∞) × U (∞) on U (∞).
The measure χ on U was described by G. Olshanski [10] and Y. Neretin [9]. Whereas, the notion U relates to D. Pickrell's space of virtual Grassmannian [11]. Problems of the Hilbert-Schmidt infinitedimensional analyticity were considered by T.A.W. Dwyer III in [3]. Hardy spaces of analytic complex functions were investigated in the works [2,8,12] and others. Spaces of integrable functions with respect to measures, invariant under infinite-dimensional groups, have been applied in the theory of determinantal point stochastic processes [1].
The paper presents following results. In Theorem 3.2, we describe an orthogonal basis in H 2 χ of power functions, indexed by means of Yang diagrams. Using this basis, in Theorem 4.2 the reproducing kernel of H 2 χ is calculated. Farther on, we define a dense embedding J : Γ H 2 χ of the symmetric Fock space Γ, defined via the Hilbert space E, which equips H 2 χ by a Hilbert-Schmidt analytic infinite-dimensional complex structure. By means of J, we establish in Theorem 6.3 an integral formula for analytic extensions of functions from H 2 χ on the open unit ball B ⊂ E. Weighted radial boundary values of these analytic extensions on B is described in Theorem 7.1.

Background on invariant measure
Let U (m) (m ∈ N) be the group of unitary (m × m)-matrices with the unit ½ m . We endow the infinite-dimensional unitary groups U (∞) with the inductive topology under embeddings U (m) U (∞) that to any u m ∈ U (m) assigns the matrix u m 0 0 ½ ∈ U (∞). Over U (∞) is defined the right action ). Following [9,10], every matrix u m ∈ U (m) with m > 1 we can write, as u m = z m−1 a b t so that z m−1 ∈ U (m − 1) and t ∈ C.
It was proven that the Livšic-type mapping (which is not a group homomorphism) from U (m) onto U (m− 1) 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 to every u m ∈ U (m) assigns the stabilizing sequence u = (u i ) i∈N (see [10, n.4]) so that Thus, the projections π m : is continuous and surjective [10,Lem 3.11]. Let U ′ = lim ← − U ′ (m) be taken with respect to these restrictions π m m−1 | U ′ (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 [10,Def 4.5]). We endow every group U (m) by the probability Haar measure χ m . As is known [9, Thm 1.6], the pullback of the measure χ m−1 on U (m − 1) under π m m−1 coincides with the measure χ m on U (m), i.e., (2.4) χ m−1 • π m m−1 = χ m for all m ∈ N. Following [10,Lem 4.8], [9, n.3.1], via of the Kolmogorov consistent theorem, we define on U the probability measure χ, which is defined to be the projective limit under the mapping (2.2), i.e., A complex function on U we call cylindrical if it has the form (2.8) The closed complex linear hull of all χ-square-integrable functions (2.6) endowed with the norm

Hardy spaces
Throughout E is a separable complex Hilbert space with an orthonormal basis {e k : k ∈ N} and the scalar product · | · , and the norm · = · | · 1/2 . In what follows, B = {x ∈ E : x < 1} stands to the unit open ball. For any element x ∈ E the following Fourier decomposition holds, Let E ⊗n stand to the complete nth tensor power of E endowed with for all x 1 ⊗ . . . ⊗ x n , y 1i ⊗ . . . ⊗ y ni ∈ E ⊗n with x ti , y ti ∈ E (t = 1, . . . , n) and for all finite sums ψ = be a partition of n ∈ N such that λ k means multiplicity of i k in (i 1 , . . . , i n ). All partitions λ can be indexed by vectors ı = (ı 1 , . . . , ı m ) consisting of cosets . . λ m > 0 and |λ| = n, where |λ| := λ 1 + . . . + λ m . By ℓ(λ) = m we denote the length of λ defined as the number of rows in λ. Let Y stand to all Young diagrams and Y n := {λ ∈ Y : |λ| = n}. Assume that Y includes the empty partition ∅ = (0, 0, . . .).
As σ : {1, . . . , n} −→ {σ(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 As is well known, the following systems of symmetric tensors e ⊙λ form orthogonal bases in E ⊙n and Γ, respectively, with the norms Hence, any element ψ ∈ Γ has the Fourier expansion For every ψ ∈ E ⊙n , we can uniquely define, so-called, the Hilbert-Schmidt n-homogenous polynomial In fact, the polarization formula for symmetric tensor products (see [5, n.1.5]) . . , e ım } and S ı := {x ∈ E ı : x = 1}. The symbol E ⊗n ı (resp., E ⊙n ı ) means the nth (resp., symmetric) tensor powers of E ı . Given an index ı ∈ N m * , we assign the appropriate S ı -valued mapping defined via the surjective projection π m : U ∋ u −→ π m (u) ∈ U (m). Consider the corresponding system of cylindrical Borel functions where ε ı k := e * ı k • ζ ı . Using ζ ı , we may define the E ⊙n ı -valued Borel mapping ζ ⊗n The following assertion is a direct consequence of [7,Lem 3].
having continuous restriction to U ′ . In particular, to every e ⊙λ ı ∈ e ⊙Yn corresponds in L ∞ χ the cylindrical function in the variable u ∈ U, Lemma 3.1 straightway implies that the system of symmetric tensors e ⊙λ ı ∈ e ⊙Y , indexed by diagrams λ = (λ 1 , . . . , λ m ) ∈ Y, uniquely defines the appropriate system of χ-essentially bounded cylindrical functions in the variable u ∈ U, possessing continuous restrictions to U ′ . Evidently, ε ∅ ı ≡ 1. Theorem 3.2. For any ı ∈ N m * and ψ, φ ∈ E ⊙n ı , the following equality holds, As a consequence, the system of cylindrical functions for all v ∈ U (m) and u ∈ U. Using (2.8), we have for all ψ, φ ∈ E ⊙n ı , where the interior integral with the Haar measures χ m are independent of π m (u) ∈ U (m). It is clear that ż for all u ∈ U. Hence, the corresponding sesquilinear form in (3.8) is continuous on E ⊙n ı . Thus, there exists a linear bounded operator A over E ⊙n ı such that Show that A commutes with all operators w ⊗n ∈ L (E ⊙n ı ) : w ∈ U (m) acting as w ⊗n x ⊗n = (wx) ⊗n , (x ∈ E ı ). Invariant properties (2.7) of χ m under the right action (2.3) yields where w −1 coincides with the hermitian adjoint matrix of w. Hence, the equality Let us suppose, on the contrary, that there is ψ ∈ E ⊙n ı such that the equality (we ı1 ) ⊗n | ψ = 0 with we ı1 ∈ S ı holds for all w ∈ U (m). The group U (m) acts surjectivity on S ı . Hence, by n-homogeneity, we obtain x ⊗n | ψ = 0 for all elements x ∈ E ı . Applying the polarization formula (3.4), we get ψ = 0. Hence, (3.10) is irreducible.
Thus, we can apply to (3.10) the Schur Lemma [6, Thm 21.30]: a non-zero 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 (3.9), is proportional to the identity operator on E ⊙n ı i.e.
In particular, the subsystem of cylindrical functions ε λ ı with a fixed ı ∈ N m * is orthogonal in L 2 χ , since the corresponding system of tensors e ⊙λ ı indexed by λ ∈ Y n with ℓ(λ) = m forms an orthogonal basis in E ⊙n ı . Remains to note that the set of all indices ı = (ı 1 , . . . , ı m ) ∈ N m * with arbitrary m = ℓ(λ) of the system ε Y is directed regarding 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 a n!/λ! in (3.11). As a result, we obtain The well known formula [13, n.1.4.9] for the unitary m-dimensional group gives Using the last two formulas, we have Combining (3.8) and (3.12), we get (3.6) and, as a consequence, (3.7).

Reproducing kernels
Let us construct the reproducing kernel of H 2 χ . We refer to [14] regarding the reproducing kernels.
Lemma 4.1. The reproducing kernel of H 2,n χ , endowed with the L 2 χ -norm, is for all u, v ∈ U and m = ℓ(λ).
Proof. Note that h 0 ≡ 1. Let us decompose any element ζ ı (u) ∈ S ı in the Fourier sum ζ ı (u) = ř m k=1 e ı k ζ ı (u) | e ı k . The tensor multinomial theorem yields the following Fourier decomposition in terms of the orthogonal basis in E ⊙n ı , Using the formula (3.2), we obtain Multiplying the both sides by n+m−1 n and summing over (m, ı) ∈ N × N m * with a fixed n ∈ N, we get (4.1).
Via Theorem 3.1 the system ε Yn forms an orthogonal basis in H 2,n χ . Hence, applying (4.1), we have i.e., the kernel (4.1) operates as the identity mapping in H 2,n χ . Consequently, the equality holds. Thus, the kernel (4.1) is reproducing in H 2,n χ .
Let us consider the following complex-valued kernel Theorem 4.2. The kernel h is reproducing in the Hardy space H 2 χ and the following decomposition holds, Proof. As is well known (see e.g. [13, n.1.4.10]), for any m ∈ N Using this equality and summing (4.1) over n ∈ Z + , we obtain where m := ℓ(λ). Hence, (4.3) holds. Every element f ∈ H 2 χ decomposes to the L 2 χ -convergent orthogonal series f = ř n f n , where f n ∈ H 2,n χ is the orthogonal projection of f in the decomposition (3.13). Observing that the L 2 χ -orthogonal property h k (·, u) ⊥ f n (·) with n = / k holds by virtue of Corollary 3.4, we obtain ż for all v ∈ U and f ∈ H 2 χ . Hence, the kernel (4.3) is reproducing in H 2 χ .

Integral formulas
Consider the dense embedding J and its adjoint mapping J * , acted as J : Γ H 2 χ , J * : H 2 χ −→ Γ, respectively, that are uniquely defined by change of orthogonal bases while keeping an appropriate orthogonal decomposition. Namely, we assign to every ψ ∈ Γ the function Jψ ∈ H 2 χ in such way that where in the left side is the Fourier decomposition of ψ ∈ Γ. Note that the equality Je ⊙λ Lemma 6.1. The operators J and J * are contractive, i.e., Proof. The equalities (3.2) and (3.7) immediately yield By Theorem 3.2 the series (6.1) are orthogonal, so in virtue of (6.3). For the adjoint operator J * it is the same.
In what follows, we assign to any x ∈ B the vector-valued functions Proof. The first assertion is a consequence of (6.3). Applying J to the Fourier decompositions (3.1) and (5.4), we obtain Taking into account (6.2), we have Hence, the function (1 − x J ) −1 with x ∈ B has values in L 2 χ .
In particular, the following series with a fixed n ∈ N, , is absolutely convergent in L 2 χ for all x ∈ E. Theorem 6.3. For any f = ř n f n ∈ H 2 χ with f n ∈ H 2,n χ the complex function belongs to the Hilbert space H 2 on B and it has the integral representation with the Taylor coefficients at the origin, Proof. Consider the Fourier decomposition with respect to the basis ε Y , wheref (λ,ı) are Fourier coefficients. Substituting coefficients to (J * f ) * , as well as, using the orthogonality together with (5.4) and (6.4), we have Similarly, using (6.5), we obtain Hence, (6.6) holds. Taking into account (3.13) and (6.8), we have Hence, the functions (6.7) coincide with the Taylor coefficients of the analytic function (J * f ) * at the origin, that are uniquely defined on B.

Weighted radial boundary values
Let us consider the functions f r := ř n r n f n ∈ H 2 χ for any f = ř n f n ∈ H 2 χ with f n ∈ H 2,n χ and r ∈ [0, 1]. In accordance with Theorem 6.3, every function (J * f r ) * ∈ H 2 has the following analytic extension on B, Taking into account the anti-linear isometries (5.2), we obtain Moreover, the following equality holds, Proof. Lemma 6.1 implies that the operator J • J * is contractive. Moreover, f k ⊥ f n as n = / k in H 2 χ by decomposition (3.13). It follows that