Sequences of Powers of Toeplitz Operators on the Bergman Space

Abstract

We consider Toeplitz operators Tu with symbol u on the Bergman space of the unit ball, and then study the convergences and summability for the sequences of powers of Toeplitz operators. We first charactreize analytic symbols φ for which the sequence $$T_\varphi ^{*k}$$ f or $$T_\varphi ^k$$ f converges to 0 or ∞ as k → ∞ in norm for every nonzero Bergman function f. Also, we characterize analytic symbols φ for which the norm of such a sequence is summable or not summable. We also study the corresponding problems on an infinite direct sum of Bergman spaces as a generalization of our result.