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

This paper studies the compressed shift operator $S_z$ on the Hardy space over the bidisk via the geometric approach. We calculate the spectrum and essential spectrum of $S_z$ on the Beurling type quotient modules induced by rational inner functions, and give a complete characterization for $S_z^*$ 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 $S_z^*$ 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 $S_z$ on certain polynomial quotient modules.