Czechoslovak Mathematical Journal, Vol. 71, No. 4, pp. 1173-1188, 2021
Schatten class generalized Toeplitz operators on the Bergman space
Chunxu Xu, Tao Yu
Received August 5, 2020. Published online June 17, 2021.
Abstract: Let $\mu$ be a finite positive measure on the unit disk and let $j\geq1$ 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 $1\leq p<\infty$ 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 $T_{\varphi_1}^{(j)}\cdots T_{\varphi_n}^{(j)}$.
Affiliations: Chunxu Xu, Tao Yu (corresponding author), School of Mathematical Sciences, Dalian University of Technology, No. 2 Linggong Road, Dalian 116024, P. R. China, e-mail: cxxu@mail.dlut.edu.cn, tyu@dlut.edu.cn
