A geometric approach to the compressed shift operator on the Hardy space over the bidisk

Abstract

This paper studies the compressed shift operator Sz on the Hardy space over the bidisk via the geometric approach. We calculate the spectrum and essential spectrum of Sz on the Beurling type quotient modules induced by rational inner functions, and give a complete characterization for Sz⁎ to be a Cowen-Douglas operator. Then we extend the concept of Cowen-Douglas operator to be the generalized Cowen-Douglas operator, and show that Sz⁎ is a generalized Cowen-Douglas operator. Moreover, we establish the connection between the reducibility of the Hermitian holomorphic vector bundle induced by kernel spaces and the reducibility of the generalized Cowen-Douglas operator. By using the geometric approach, we study the reducing subspaces of Sz on certain polynomial quotient modules.