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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 810735, 7 pages
Research Article

Hardy-Type Space Associated with an Infinite-Dimensional Unitary Matrix Group

Institute of Mathematics, University of Rzeszów, 16A Rejtana Street, 35-310 Rzeszów, Poland

Received 23 March 2013; Accepted 18 June 2013

Academic Editor: Qing-Wen Wang

Copyright © 2013 Oleh Lopushansky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We investigate an orthogonal system of the homogenous Hilbert-Schmidt polynomials with respect to a probability measure which is invariant under the right action of an infinite-dimensional unitary matrix group. With the help of this system, a corresponding Hardy-type space of square-integrable complex functions is described. An antilinear isomorphism between the Hardy-type space and an associated symmetric Fock space is established.

1. Introduction

We investigate an orthogonal system of the Hilbert-Schmidt polynomials in the space of square-integrable complex functions on the projective limit of unitary -dimensional matrix groups , called the space of virtual unitary matrices and endowed with the projective limit measure of the probability Haar measures on . The measure on the space is invariant under the right action of the infinite-dimensional unitary group, where.

The space of virtual unitary matriceswas studied by Neretin [1] and Olshanski [2]. This notion relates to D. Pickrell’s space of virtual Grassmannian [3] and to Kerov, Olshanski, and Vershik’s space of virtual permutations [4]. Various spaces of integrable functions with respect to measures that are invariant under infinite-dimensional groups have been widely applied in stochastic processes [5], infinite-dimensional probability [6, 7], complex analysis [8], and so forth.

The main results of the present paper are Theorems 67 that describe a Hardy-type subspacespanned by the finite type homogenous Hilbert-Schmidt polynomials that are generated by an associated symmetric Fock space.

2. Preliminaries

We consider the following infinite-dimensional unitary matrix groups: whereis the group of unitary-matrices which is identified with the subgroup infixing theth basis vector. In other words,is the group of infinite unitary matrices with finitely many matrix entriesdistinct from. We equip every group with the probability Haar measure.

Following [1, 2], every matrix with, we write in the following block matrix form: corresponding to the partition so that and . Over the group (resp., ) the right action is well defined: where belongs to (resp., to ) and belongs to (resp., to ). In [1, Proposition ], [2, Lemma 3.1], it was proven that the following Livšic-type mapping: such that (which is not a group homomorphism) is Borel and surjective onto and commutes with the right action of.

As is known [1, Theorem 1.6], the pullback of the probability Haar measureonunder the mappingis the probability Haar measureon, that is,

Let be the subset of unitary matrices which do not have, as an eigenvalue. Then, is open in, and the complement is a-negligible set. Moreover (see [2, Lemma 3.11]), the mapping is continuous and surjective.

Consider the projective limits, taken with respect to the surjective Borel projectionsand their continuous restrictions, respectively, called the spaces of virtual unitary matrices. Notice thatis a Borel subset in the Cartesian product endowed with the product topology, because all mappingare Borel. Moreover, the canonical projections such that, are surjective by surjectivity ofand.

Following [2, Lemma ], [1, Section 3.1], with the help of the Kolmogorov consistent theorem, we uniquely define a probability measureonas the projective limit under the mapping (6), which satisfies the equalityfor all. On, the measureis zero, becauseis zero onfor all.

Using (3), right action of the groupon the space of virtual unitary matricescan be defined (see [2, Definition 4.5]) as follows: where is so large that .

The canonical dense embedding to any elementassigns the unique sequence , such that whereis the unit in. So, the imageconsists of stabilizing sequences in(see [2, Section 4]).

3. Invariant Probability Measure

In what follows, we will endow the space of virtual unitary matriceswith the measure. A complex function onis called cylindrical [2, Definition 4.5] if it has the following form: for a certainand a certain complex functionon.

Any continuous bounded functiononhas a unique-essentially bounded extension on, because the setis-negligible. Therefore, if the functionin the definition (13) is continuous and bounded, then the corresponding cylindrical functionis essentially bounded.

By, we denote closure of the algebraic hull of all cylindrical-essentially bounded functions (13) with respect to the following norm:

Lemma 1. The measureonis a Radon probability measure such that for alland. For any compact setthe following equality holds:

Proof. Recall the Prohorov criterion, which is adapted to our notation (see [9, Chapter IX.4.2, Theorem ] or [6, Theorem 6]): there exists a Radon probability measureonsuch that if and only if for everythere exists a compact setinsuch that the following inequality holds; in this case, is uniquely determined by means of the formula, whereis a compact set in.
Letbe a compact set with a fixed . Putting, we have On the other hand, if we put, then via (6), As a consequence, the compact setin, generated by a compact setwith the help of mappings, satisfies the following condition:
The probability Haar measureis regular on, and the complementis a negligible set. Hence, ifruns over all compact sets in, then Therefore, for everythere exists a compact setsuch that. From (21), it follows that for everythe compact setsatisfies the hypothesis of Prohorov’s criterion: So, in view of this criterion, there exists a unique Radon probability measureonwhich satisfies the condition (17). However, on the projective limits, there exists a unique-invariant Radon measure, determined by the equality (15). Using the uniqueness property of projective limits, we obtain. The measureonis defined to be zero, becauseis zero on.
As a consequence of (21), we obtain (16), because
As is known [1, Proposition 3.2], the measureis-invariant under the right actions (11) on the space. Hence, for every, the equality (15) holds.

4. Shift Groups

Consider that in the space, the group of shifts is generated by the right action ofover. Choosing instead ofa compact subgroup or the compact subgroups we obtain the corresponding subgroups of shiftswith elementsor with elements and, respectively. Here, means the unit element in.

Lemma 2. For anythe following equalities: withorhold.

Proof. For any, the function is integrable on the Cartesian product . By the Fubini theorem, we obtain
This equality yields the required formula (27), because the internal integral on the right-hand side is independent ofand. In turn, putting instead ofthe subgroupsand , we obtain equalities (28).

5. The Homogeneous Hilbert-Schmidt Polynomials

Consider the countable orthogonal Hilbertian sum with the scalar product, where every coordinateis identified with its image under the embedding.

Let stand for the complete th tensor power of the Hilbert subspace , endowed with the Hilbertian scalar product and norm, respectively, where , withfor all and denotes a finite sum. Put . We use the following short denotation:

Replacing the spaceby the subspace, we similarly define the tensor product. There is the unitary embedding . If , then .

For any finite sumfrom the space(or), we can to define the finite type -homogeneous Hilbert-Schmidt polynomials:

Consider the canonical orthonormal bases: where.

Ifruns over all-elements permutations, then the symmetricth tensor poweris defined to be a codomain of the symmetrization mapping: which is an orthogonal projector. Similarly, the symmetricth tensor powercan be defined. Clearly, is a closed subspace in.

Given a pair of numbers, we consider the-fold tensor power of the canonical mapping , where . If, we putfor alland. The mapping (36) is Borel and has a continuous restriction to, becausehas the same property (see Section 2).

Let be an arbitrary fixed element such that . Then, . Using the mapping (36), we can write To any -homogeneous Hilbert-Schmidt polynomial (33), there corresponds the function of the variable. Any cylindrical function of the formhas a continuous bounded restriction to. Therefore, it is-essentially bounded on, because is a-negligible set. Consequently,and is continuous and bounded.

Definition 3. We defineto be the space of all functionsof the variable, determined by the finite type-homogeneous Hilbert-Schmidt polynomials (33).

Lemma 4. For any element such that the set coincides with the unit sphere in. As a consequence, the one-to-one antilinear corresponding Holds, and any function is independent of the choice of an element .

Proof. Suppose, on the contrary, that there is an elementsuch thatfor all with. The mapping is surjective by surjectivity of the mapping(see [2, Lemma 3.1]). Hence, the set coincides with the unit sphere in and is independent on the choice of an element . By -homogeneity, we have for all .
Apply the following polarization formula for symmetric tensor products (see, e.g., [10, Section 1.5]): with, which is valid for all. It follows that for all elements. Hence, , because the subset of all elementsis total in. As a consequence, the subset is also total in. It immediately yields the correspondence (40).

Consider the symmetric Fock space and its closed subspace , where We will use the following notations: As is well known (see, e.g., [11]), the system of symmetric tensor elements, indexed by the set , forms an orthogonal basis in the subspace We will also use the following notations: Then, the system of symmetric tensor elements with a fixed, indexed by the setsand, forms an orthogonal basis in the subspace. Thus, the system forms an orthogonal basis in the symmetric Fock space .

By virtue of the one-to-one mapping (40), the system of symmetric tensor elementsuniquely defines the following corresponding system: of the following -integrable cylindrical functions: of the variable, where we take. Consider the system of functions of the variable, generated by the system of symmetric tensor elements. All these functions belong to the spaceby their definition. Denote

6. The Hardy-Type Space

Letbe the space of square-integrable complex functionson the space of virtual matrices. Since is a probability measure, the embeddingholds and

Denote bythe-closure of complex linear spans of the subsystem. As is well known (see, e.g., [12, Theorem ]), the spaceis isomorphic to the classic Hardy spaceof analytic complex functions on the open unit ball . Therefore, the following more general definition seems natural (see, also [8]).

Definition 5. The Hardy-type spaceon the space of virtual unitary matricesis defined to be the-closure of the complex linear span of the system.

Theorem 6. The systemof all functionswith, such that as, forms an orthogonal basis in the Hardy-type spaceswith norms

Proof. If, then from (28), it follows that So, in the spaceiffor all indices.
Let and for definiteness. If the elements and are different, then there exists a subindex in the block-index such that , where . The formula (28) implies that for the group of shifts generated by elements with the subindex, Hence,in.
Let nowand. If, then. Hence, there exists a sub-indexin the block-index such that. Similarly as previous mentioned, applying the formula (28) to the group of shiftsgenerated by elementswith the sub-index, we get Hence, in this case alsounder the measure.
Letand. Using (11) and (52), we have However, the previous integral with the Haar measureis independent of. It follows that by the well-known formula [12, Section ]. Using the formula (27)-times for, we get because. It follows that for all.

As is known (see, e.g., [11]), the systemof symmetric tensorswith a fixedforms an orthogonal basis in the symmetric Fock spacewith norms. Similarly, the systemof symmetric tensorswith all, such thatas, forms an orthogonal basis in the symmetric Fock spacewith norms .

Combining Lemma 4, Theorem 6, and [12, Theorem ], we obtain the following.

Theorem 7. Antilinear extensions of the one-to-one mappings between the orthonormal bases uniquely define the corresponding anti-linear isometric isomorphisms

Reasoning by analogy with [8, Proposition 6.1 and Theorem 7.1], it is easy to show that the Hardy space possesses the reproducing kernel of a Cauchy type with, where the sumis over all indicessuch that . As a consequence, the integral representation of any function, gives a unique analytic extension in the complex variablefor all elementssuch that


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