[COLLOQUIUM] A GEOMETRIC APPROACH TO THE COMPRESSED SHIFT OPERATORS ON HARDY SPACE OVER THE BIDSK

Due to the relation with the invariant subspace problem, the structure of compressed shift operator is one of the important problems in operator theory.  This talk focuses on the reducibility of the compressed shift operator on the Hardy space over the bidisk via the geometric approach. We introduce Alger reducibility and pure isometry reducibility, calculate the spectrum and essential spectrum of on the Beurling type quotient modules induced by rational inner functions, and give a complete characterization for to be a Cowen-Douglas operator. Then we extend the notion of Cowen-Douglas operator to be the generalized Cowen-Douglas operator, and the connection between the reducibility of and generalized Cowen-Douglas operator are established. By using the geometric approach, we also study the reducing subspaces of on certain polynomial quotient modules.