Hankel operators and atomic decompositions in vector-valued Bergman spaces

Abstract

This thesis consists of the following three papers

Paper I. Hankel operators on Bergman spaces and similarity to contractions.

In this paper we consider Foguel-Hankel operators on vector-valued Bergman spaces. Such operators defined on Hardy spaces play a central role in the famous example by Pisier of a polynomially bounded operator which is not similar to a contraction. On Bergman spaces we encounter a completely different behaviour; power boundedness, polynomial boundedness and similarity to a contraction are all equivalent for this class of operators.

Paper II. Weak product decompositions and Hankel operators on vector-valued Bergman spaces.

We obtain weak product decomposition theorems, which represent the Bergman space analogues to Sarason's theorem for operator-valued Hardy spaces, respectively, to the Ferguson-Lacey theorem for Hardy spaces on product domains. We also characterize the compact Hankel operators on vector-valued Bergman spaces.

Paper III. Discretizations of integral operators and atomic decompositions in vector-valued Bergman spaces.

We prove a general atomic decomposition theorem for weighted vector-valued Bergman spaces, which has applications to duality problems and to the study of compact Toeplitz type operator