Schatten class generalized Toeplitz operators on the Bergman space

@article{Xu2021,
abstract = {Let $\mu $ be a finite positive measure on the unit disk and let $j\ge 1$ be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator $T_\{\mu \}^\{(j)\}$ to be bounded or compact. We first give a necessary and sufficient condition for $T_\{\mu \}^\{(j)\}$ to be in the Schatten $p$-class for $1\le p<\infty $ on the Bergman space $A^\{2\}$, and then give a sufficient condition for $T_\{\mu \}^\{(j)\}$ to be in the Schatten $p$-class $(0<p<1)$ on $A^\{2\}$. We also discuss the generalized Toeplitz operators with general bounded symbols. If $\varphi \in L^\{\infty \}(D, \{\rm d\}A)$ and $1<p<\infty $, we define the generalized Toeplitz operator $T_\{\varphi \}^\{(j)\}$ on the Bergman space $A^p$ and characterize the compactness of the finite sum of operators of the form $T_\{\varphi _1\}^\{(j)\}\cdots T_\{\varphi _n\}^\{(j)\}$.},
author = {Xu, Chunxu, Yu, Tao},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized Toeplitz operator; Schatten class; compactness; Bergman space; Berezin transform},
language = {eng},
number = {4},
pages = {1173-1188},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Schatten class generalized Toeplitz operators on the Bergman space},
url = {http://eudml.org/doc/298263},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Xu, Chunxu
AU - Yu, Tao
TI - Schatten class generalized Toeplitz operators on the Bergman space
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 4
SP - 1173
EP - 1188
AB - Let $\mu $ be a finite positive measure on the unit disk and let $j\ge 1$ be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator $T_{\mu }^{(j)}$ to be bounded or compact. We first give a necessary and sufficient condition for $T_{\mu }^{(j)}$ to be in the Schatten $p$-class for $1\le p<\infty $ on the Bergman space $A^{2}$, and then give a sufficient condition for $T_{\mu }^{(j)}$ to be in the Schatten $p$-class $(0<p<1)$ on $A^{2}$. We also discuss the generalized Toeplitz operators with general bounded symbols. If $\varphi \in L^{\infty }(D, {\rm d}A)$ and $1<p<\infty $, we define the generalized Toeplitz operator $T_{\varphi }^{(j)}$ on the Bergman space $A^p$ and characterize the compactness of the finite sum of operators of the form $T_{\varphi _1}^{(j)}\cdots T_{\varphi _n}^{(j)}$.
LA - eng
KW - generalized Toeplitz operator; Schatten class; compactness; Bergman space; Berezin transform
UR - http://eudml.org/doc/298263
ER -