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Keywords:
generalized Toeplitz operator; Schatten class; compactness; Bergman space; Berezin transform
generalized Toeplitz operator; Schatten class; compactness; Bergman space; Berezin transform
Summary:
Let $\mu $ be a finite positive measure on the unit disk and let $j\geq 1$ 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\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 $T_{\varphi _1}^{(j)}\cdots T_{\varphi _n}^{(j)}$.
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