Abstract
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.
Highlights
We investigate an orthogonal system of the Hilbert-Schmidt polynomials in the space L2χ of square-integrable complex functions on the projective limit U = li←mU(m) of unitary (m × m)-dimensional matrix groups U(m) (m ∈ N), called the space of virtual unitary matrices and endowed with the projective limit measure χ = li←mχm of the probability Haar measures χm on U(m)
The main results of the present paper are Theorems 67 that describe a Hardy-type subspace H2χ ⊂ L2χ spanned by the finite type homogenous Hilbert-Schmidt polynomials that are generated by an associated symmetric Fock space
[1, Proposition 0.1], [2, Lemma 3.1], it was proven that the following Livsic-type mapping: πmm−1 : U (m) ∋ um → um−1 ∈ U (m − 1), (4)
Summary
We investigate an orthogonal system of the Hilbert-Schmidt polynomials in the space L2χ of square-integrable complex functions on the projective limit U = li←mU(m) of unitary (m × m)-dimensional matrix groups U(m) (m ∈ N), called the space of virtual unitary matrices and endowed with the projective limit measure χ = li←mχm of the probability Haar measures χm on U(m). The space of virtual unitary matrices U was studied by Neretin [1] and Olshanski [2]. Various spaces of integrable functions with respect to measures that are invariant under infinite-dimensional groups have been widely applied in stochastic processes [5], infinitedimensional probability [6, 7], complex analysis [8], and so forth. The main results of the present paper are Theorems 67 that describe a Hardy-type subspace H2χ ⊂ L2χ spanned by the finite type homogenous Hilbert-Schmidt polynomials that are generated by an associated symmetric Fock space
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