English

Abstract

Listen

Translate article

Let μ be a finite positive measure on the unit disk and let j ⩾ 1 be an integer. D. Suarez (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 Ap and characterize the compactness of the finite sum of operators of the form $$T_{{\varphi _1}}^{(j)} \ldots T_{{\varphi _n}}^{(j)}$$ .