On irreducibility of a certain class of homogeneous operators obtained from quotient modules

TL;DR

This paper investigates the irreducibility of certain homogeneous operators derived from quotient modules of Hilbert modules over complex domains, revealing conditions under which these operators are reducible and identifying their components.

Contribution

It establishes the homogeneity of compressed multiplication operators on quotient modules and provides explicit examples of reducibility, linking components to Generalized Wilkins' operators.

Findings

Compressed operators are homogeneous under specific automorphism subgroups.

These operators can be reducible even if originating operators are irreducible.

Irreducible components are characterized as Generalized Wilkins' operators.

Abstract

Let $ \Omega \subset \mathbb{C}^m $ be an open, connected and bounded set and $\mathcal{A}(\Omega)$ be a function algebra of holomorphic functions on $\Omega$. Suppose that $\mathcal{M}_q$ is the quotient Hilbert module obtained from a submodule of functions in a Hilbert module $\mathcal{M}$ vanishing to order $k$ along a smooth irreducible complex analytic set $\mathcal{Z}\subset\Omega$ of codimension at least $2$. In this article, we prove that the compression of the multiplication operators onto $\mathcal{M}_q$ is homogeneous with respect to a suitable subgroup of the automorphism group Aut$(\Omega)$ of $\Omega$ depending upon a subgroup $G$ of Aut$(\Omega)$ whenever the tuple of multiplication operators on $\mathcal{M}$ is homogeneous with respect to $G$ and both $\mathcal{M}$ as well as $\mathcal{M}_q$ are in the Cowen-Douglas class. We show that these compression of multiplication operators might be reducible even if the tuple of multiplication operators on $\mathcal{M}$ is irreducible by exhibiting a concrete example. Moreover, the irreducible components of these reducible operators are identified as Generalized Wilkins' operators.