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In 1997 Pták defined generalized Hankel operators as follows: Given two contractions and , an operator is said to be a generalized Hankel operator if and satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of and . This approach, call it (P), contrasts with a previous one developed by Pták and Vrbová in 1988, call it (PV), based on the existence of a previously defined generalized Toeplitz operator. There seemed to be a strong but somewhat...
In this paper we study spaces of holomorphic functions on the right half-plane R, that we denote by Mpω, whose growth conditions are given in terms of a translation invariant measure ω on the closed half-plane R. Such a measure has the form ω = ν ⊗ m, where m is the Lebesgue measure on R and ν is a regular Borel measure on [0, +∞). We call these spaces generalized Hardy–Bergman spaces on the half-plane R. We study in particular the case of ν purely atomic, with point masses on an arithmetic progression...
We establish an inversion formula for the M. M. Djrbashian A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain , . We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the -Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of . We build an anti-holomorphic embedding of in the complex projective Hilbert space and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability amplitudes....
It is shown that in the Dirichlet space , two invariant subspaces ℳ ₁, ℳ ₂ of the Dirichlet shift are unitarily equivalent only if ℳ ₁ = ℳ ₂.
Let f, g be in the analytic function ring Hol(𝔻) over the unit disk 𝔻. We say that f ⪯ g if there exist M > 0 and 0 < r < 1 such that |f(z)| ≤ M|g(z)| whenever r < |z| < 1. Let X be a Hilbert space contained in Hol(𝔻). Then X is called an ordered Hilbert space if f ⪯ g and g ∈ X imply f ∈ X. In this note, we mainly study the connection between an ordered analytic Hilbert space and its reproducing kernel. We also consider when an invariant subspace of the whole space X is similar...
We consider positive definite kernels which are invariant under a multiplier and an action of a semigroup with involution, and construct the associated projective isometric representation on a Hilbert C*-module. We introduce the notion of C*-valued Hilbert-Schmidt kernels associated with two sequences and construct the corresponding reproducing Hilbert C*-module. We also discuss projective invariance of Hilbert-Schmidt kernels. We prove that the range of a convolution type operator associated with...
The purpose of this paper is to exhibit a connection between the Hermitian solutions of matrix Riccati equations and a class of finite dimensional reproducing kernel Krein spaces. This connection is then exploited to obtain minimal factorizations of rational matrix valued functions that are J-unitary on the imaginary axis in a natural way.
We study the bases and frames of reproducing kernels in the model subspaces
of the Hardy class in the upper half-plane. The main problem under consideration is the stability of a basis of reproducing kernels
under “small” perturbations of the points . We propose an approach to this problem based on the recently obtained estimates of derivatives in the spaces
and produce estimates of admissible perturbations generalizing certain results of W.S. Cohn and E. Fricain.
Let be a Hermitian symmetric space of the noncompact type and let be a discrete series representation of holomorphically induced from a unitary character of . Following an idea of Figueroa, Gracia-Bondìa and Vàrilly, we construct a Stratonovich-Weyl correspondence for the triple by a suitable modification of the Berezin calculus on . We extend the corresponding Berezin transform to a class of functions on which contains the Berezin symbol of for in the Lie algebra of . This allows...
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