Essentially normal Hilbert modules and K-homology III: Homogenous quotient modules of Hardy modules on the bidisk

Abstract

In this paper, we study the homogenous quotient modules of the Hardy module on the bidisk. The essential normality of the homogenous quotient modules is completely characterized. We also describe the essential spectrum for a general quotient module. The paper also considers Khomology invariant defined in the case of the homogenous quotient modules on the bidisk.

Beurling type quotient modules over the bidisk and boundary representations

“…One motivation is from an attempt to understand operator theory, function theory and related geometric analysis over the bidisk. There is a large literature concerning the study of essential normality over the bidisk, see [12,11,17,18,23,25,28], here we have made no attempt to compile a comprehensive list of references.…”

Quotient modules for some Hilbert modules over the bidisk

“…An important fact about the Hardy module over the bidisk is that the multiplication operators are isometric which makes the Hardy module very special. But in our setting, this property does not hold necessarily, the proof here is different from that in [20,21,23].…”

“…[20,21,23] In Section 2, we will prove the essential normality of some type of quotient modules of a large variety of natural reproducing kernel Hilbert modules over the bidisk including the Hardy module. If one consider the Hardy module, there are some very interesting results, we refer the reader to [20,21,23]. An important fact about the Hardy module over the bidisk is that the multiplication operators are isometric which makes the Hardy module very special.…”

An index formula for the two variable Jordan block

“…The compactness of L(0)| N and R(0)| N has a closed relationship with the essential normality of N ; for example, see [GW2]. We will characterize the compactness of L(0) on Beurling type quotient module completely.…”