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Generalized Berezin transform and Schatten class operators on weighted Bergman spaces
Authors: Das N., Roy S.
Keywords: weighted Bergman spaces, reproducing kernel, Toeplitz operator, little Hankel operator, Schatten class operators
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Keywords: weighted Bergman spaces, reproducing kernel, Toeplitz operator, little Hankel operator, Schatten class operators
Abstract
In this paper we derive certain elementary properties of the generalized Berezin transform $B_\alpha,\; \alpha>-1$ and show that if $T$ is a bounded linear operator from the weighted Bergman space $L^{2}_a(dA_\alpha)$ into itself, and $G(w)= \left\Vert TK_w^{(\alpha)}\right\Vert^p,\; w\in \mathbb{D},\; 0 p<\infty$, then $B_0^n G$ is subharmonic for all $n\in\mathbb{N}$ and $B_0^n \phi \rightarrow \psi,$ the least harmonic majorant of $G$. Further, we describe the Schatten class characterization of a bounded linear operator $T$ in terms of the generalized Berezin transform of $T$. We have also shown that if $T\geq 0$ or $T$ is in trace class $S_1$, then $tr(T)= (\alpha+1)\displaystyle\int\limits_{\mathbb D} (B_\alpha T)(z)d\lambda(z)$ where $d\lambda(z)= \frac{dA(z)}{\left( 1-|z|^2 \right)^2}$. Further, if $A$ is compact and positive and $0NAMITA DAS
P. G. Dept. of Mathematics, Utkal University,Vani Vihar,
Bhubaneswar- 751004, Odisha, India
E-mail:
SWARUPA ROY
Silicon University, Silicon Hills, Patia, Bhubaneswar-
751024, Odisha, India.
E-mail:
P. G. Dept. of Mathematics, Utkal University,Vani Vihar,
Bhubaneswar- 751004, Odisha, India
E-mail:
SWARUPA ROY
Silicon University, Silicon Hills, Patia, Bhubaneswar-
751024, Odisha, India.
E-mail:
DOI
https://doi.org/10.56415/basm.y2025.i1.p3Fulltext
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Contents
- Generalized Berezin transform and Schatten class operators on weighted Bergman spaces
- Reverse Hardy inequalities via μ-proportional generalized fractional integral operators
- Some integrals for cosine families of bounded linear operators on some non-Archimedean Banach spaces
- About the Laplace transform on ${{L}^{1}}\left( \mathbb{R^+} \right)$
- Vertical Generalized Berger-type Metrics: A New Perspective on Harmonic Vector Fields
- On conharmonic curvature tensor of 6-dimensional planar Hermitian submanifolds of Cayley algebra
- Completeness of the factor group of a complete topological group and completeness of the factor ring of a complete topological ring by a compact normal subgroup and a compact ideal, respectively.
- Pure and Mixed Stationary Equilibria for Dynamic Positional Games on Graphs
- On 4-T-quasigroups with exactly 20 distinct parastrophes
