N_φ-type quotient modules on the torus

Structure of the quotient modules in H²(r²) 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(F ²) Q [z - φ], where φ(w) is a function in the classical Hardy space H²(r) 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(r²) ⊖ [z - φ].