SCHATTEN CLASS GENERALIZED TOEPLITZ OPERATORS ON THE BERGMAN SPACE

Let mu 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 < infinity 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 phi is an element of L-infinity(D, dA) and 1 < p < infinity, we define the generalized Toeplitz operator T-phi((j)) on the Bergman space A(p) and characterize the compactness of the finite sum of operators of the form T-phi 1((j)) . . . T-phi n((j)).

Automated summary

The study explores the properties of the generalized Toeplitz operator T-mu((j)) on the unit disk with a finite positive measure mu, discussing conditions for boundedness and compactness. It provides necessary and sufficient conditions for the operator in the Schatten p-class on the Bergman space A(2), as well as sufficient conditions for the Schatten p-class (0<p<1) on A(2). Additionally, it examines generalized Toeplitz operators with general bounded symbols and characterizes the compactness of finite sums of operators in the form of T-phi 1((j)) . . . T-phi n((j)) on the Bergman space A(p) with phi in L-infinity(D, dA) and 1<p<infinity.