Resolvent set analysis of the Bergman shift

Abstract

This paper studies the invariant subspaces of the Bergman shift using the resolvent set analysis approach introduced by Douglas and Yang. We construct invariants on the resolvent set of the Bergman shift to describe the Bergman inner functions and the inclusion relationship of invariant subspaces of the Bergman shift. We generalize the concept of power sets, originally introduced by Douglas and Yang for quasinilpotent operators, to any boundary point of the spectrum of any operator. We compute the newly defined power sets for the conjugate of a class of compression operators of the Bergman shift, and demonstrate how they reflect the structure of invariant subspaces.