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 T_z and T_w 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 operator S_z on H²⊝[z-w] is unitarily equivalent to the Bergman shift, so the Beurling type theorem holds for S_z on H²⊝[z-w]. As a generalization, we shall show that the Beurling type theorem holds for S_z on H²⊝[z-φ(w)]. Also we shall prove that the Beurling type theorem holds for the fringe operator F_w on [z-w]⊝z[z-w] and for F_z on [z-φ(w)]⊝w[z-φ(w)] if φ(0)=0.

The first author is partially supported by Grant-in-Aid for Scientific Research (No.21540166), Japan Society for the Promotion of Science.