New York Journal of Mathematics
Volume 14 (2008) 431-457


Keiji Izuchi and Rongwei Yang

Nϕ-type quotient modules on the torus

Published: September 17, 2008
Keywords: The Hardy space on the torus, quotient modules, two variable Jordan block, evaluation operators, essential reductivity
Subject: Primary 46E20; Secondary 47A13

Structure of the quotient modules in H22) is very complicated. A good understanding of some special examples will shed light on the general picture. This paper studies the so-called Nϕ-type quotient modules, namely, quotient modules of the form H22)⊝ [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 Sz and Sw, compactness of evaluation operators, and essential reductivity of H22)⊝ [z-ϕ].


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

Author information

Keiji Izuchi:
Department of Mathematics, Niigata University, Niigata, 950-2181, Japan

Rongwei Yang:
Department of Mathematics and Statistics, SUNY at Albany, Albany, NY 12047, U.S.A.