Wandering subspaces and the Beurling type theorem. II

摘要

Let H2 be the Hardy space over the bidisk.Let φ(w) be a nonconstant inner function.We denote by [z-φ(w)] the smallest invariant subspace for both operators Tz and Tw containing the function z-φ(w).Aleman, Richter and Sundberg showed that the Beurling type theorem holds for the Bergman shift on the Bergman space. It is known that the compression operatorSz on H2⊝[z-w] is unitarily equivalent to the Bergman shift,so the Beurling type theorem holds for Sz on H2⊝[z-w]. As a generalization, we shall show that the Beurling type theorem holds for Sz on H2⊝[z-φ(w)]. Also we shall prove that the Beurling type theorem holds for the fringe operatorFw on [z-w]⊝z[z-w] and for Fz on [z-φ(w)]⊝w[z-φ(w)] if φ(0)=0.