Cowen-Douglas operators and analytic continuation

TL;DR

This paper explores the relationship between Cowen-Douglas operators and analytic continuation within Banach spaces of analytic functions, revealing how invariant subspaces' spectra influence function extension properties.

Contribution

It demonstrates that under natural conditions, the left inverse of a multiplication operator is a Cowen-Douglas operator and analyzes the connection to analytic continuation.

Findings

Left inverse of multiplication operator is a Cowen-Douglas operator under certain conditions.

Analytic continuation of functions relates to the spectrum of the operator's restriction.

Provides insights into spectral properties of invariant subspaces.

Abstract

In this paper, we study certain Banach spaces of analytic functions on which a left-invertible multiplication operator acts. It turns out that, under natural conditions, its left inverse is a Cowen-Douglas operator. We investigate how the analytic continuations of functions from an invariant subspace of this Cowen-Douglas operator relate to the spectrum of its restriction to that subspace.