Structure of the quotient modules in H²(Γ²) is very complicated. A good understanding of some special examples will shed light on the general picture. This paper studies the so-called N_ϕ-type quotient modules, namely, quotient modules of the form H²(Γ²)⊝ [z-ϕ], where ϕ (w) is a function in the classical Hardy space H²(Γ) and [z-ϕ] is the submodule generated by z-ϕ (w). This type of quotient module provides good examples in many studies. A notable fact is its close connections with some classical operators, namely the Jordan block and the Bergman shift. This paper studies spectral properties of the compressions S_z and S_w, compactness of evaluation operators, and essential reductivity of H²(Γ²)⊝ [z-ϕ].

The first author is partially supported by Grant-in-Aid for Scientific Research (No.16340037), Ministry of Education, Science and Culture.