Schatten class generalized Toeplitz operators on the Bergman space

Abstract

Let μ be a finite positive measure on the unit disk and let j ⩾ 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 ⩽ p < ∞ on the Bergman space A², and then give a sufficient condition for $T_\mu ^{(j)}$ to be in the Schatten p-class (0 < p < 1) on A². We also discuss the generalized Toeplitz operators with general bounded symbols. If ϕ ∈ L^∞ (D, dA) and 1 < p < ∞, 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 $T_{{\varphi _1}}^{(j)} \ldots T_{{\varphi _n}}^{(j)}$.