M_φ-type Submodules over the Bidisk

Abstract

Let ${H^2}\left( {{\mathbb{D}^2}} \right)$ be the Hardy space over the bidisk ${\mathbb{D}^2}$, and let M_φ = [(z − φ(w))²] be the submodule generated by (z − φ(w))², where φ(w) is a function in H^∞(w). The related quotient module is denoted by ${N_\varphi } = {H^2}\left( {{\mathbb{D}^2}} \right)\Theta {M_\varphi }$. In the present paper, we study the Fredholmness of compression operators S_z, S_w on N_φ. When φ(w) is a nonconstant inner function, we prove that the Beurling type theorem holds for the fringe operator F_w on [(z − w)²] ⊖ z[(z − w)²] and the Beurling type theorem holds for the fringe operator F_z on M_φ ⊖ wM_φ if φ(0) = 0. Lastly, we study some properties of F_w on [(z − w²)²] ⊖ z[(z − w²)²].