In 1997 Pták defined generalized Hankel operators as follows: Given two contractions $T_1\in ℬ\left({\mathscr{H}}_1\right)$ and $T_2\in ℬ\left({\mathscr{H}}_2\right)$, an operator $X{\mathscr{H}}_1\to {\mathscr{H}}_2$ is said to be a generalized Hankel operator if $T_{2}X=X{T}_1^{*}$ and $X$ satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of $T_1$ and $T_2$. 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 ${\Omega }_n=\zeta \in {ℂ}^n:\varrho \left(\zeta \right)>0$, $\varrho \left(\zeta \right)=Im\left({\zeta }_1\right)-{\left|{\zeta }^{\text{'}}\right|}^2$. We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the ${\varrho }^{\alpha }$-Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of ${\Omega }_n$. We build an anti-holomorphic embedding of ${\Omega }_n$ in the complex projective Hilbert space $ℂ\mathbb{P}\left(H_{\alpha }^2\left({\Omega }_n\right)\right)$ 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 $M_z$ 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 $K_{\Theta }^2=H^2\ominus \Theta H^2$ of the Hardy class $H2$ in the upper half-plane. The main problem under consideration is the stability of a basis of reproducing kernels $k_{{\lambda }_n}\left(z\right)=(1-\overline{\Theta \left({\lambda }_n\right)}\Theta \left(z\right))/(z-{\overline{\lambda }}_n)$ under “small” perturbations of the points ${\lambda }_n$. We propose an approach to this problem based on the recently obtained estimates of derivatives in the spaces $K_{\Theta }^2$ and produce estimates of admissible perturbations generalizing certain results of W.S. Cohn and E. Fricain.

Let $M=G/K$ be a Hermitian symmetric space of the noncompact type and let $\pi$ be a discrete series representation of $G$ holomorphically induced from a unitary character of $K$. Following an idea of Figueroa, Gracia-Bondìa and Vàrilly, we construct a Stratonovich-Weyl correspondence for the triple $(G,\pi ,M)$ by a suitable modification of the Berezin calculus on $M$. We extend the corresponding Berezin transform to a class of functions on $M$ which contains the Berezin symbol of $d\pi \left(X\right)$ for $X$ in the Lie algebra $𝔤$ of $G$. This allows...