M φ -type Submodules over the Bidisk

Let $${H^2}\left( {{\mathbb{D}^2}} \right)$$ be the Hardy space over the bidisk $${\mathbb{D}^2}$$ , and let Mφ = [(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 Sz, Sw on Nφ. When φ(w) is a nonconstant inner function, we prove that the Beurling type theorem holds for the fringe operator Fw on [(z − w)2] ⊖ z[(z − w)2] and the Beurling type theorem holds for the fringe operator Fz on Mφ ⊖ wMφ if φ(0) = 0. Lastly, we study some properties of Fw on [(z − w2)2] ⊖ z[(z − w2)2].