Schatten class generalized Toeplitz operators on the Bergman space

Xu, Chunxu ; Yu, Tao

Czechoslovak Mathematical Journal, Tome 71 (2021) no. 4, pp. 1173-1188

Résumé

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 $(0$ on $A^{2}$. We also discuss the generalized Toeplitz operators with general bounded symbols. If $\varphi \in L^{\infty }(D, {\rm d}A)$ and $1$, 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 $T_{\varphi _1}^{(j)}\cdots T_{\varphi _n}^{(j)}$.

Détail

DOI : 10.21136/CMJ.2021.0336-20

Classification : 47B10, 47B35
Keywords: generalized Toeplitz operator; Schatten class; compactness; Bergman space; Berezin transform