Analytic continuation and embeddings in weighted backward shift invariant subspaces

TL;DR

This paper explores the analytic continuation properties of functions in weighted backward shift invariant subspaces in Hardy spaces, revealing how spectrum characteristics influence their extendability and embeddings, especially in relation to Toeplitz operators.

Contribution

It extends classical results on backward shift invariant subspaces to the weighted setting, analyzing spectrum effects and embedding properties in relation to Toeplitz operators.

Findings

Functions extend analytically through holes in the spectrum.

Weighted invariant subspaces embed into unweighted counterparts.

Spectrum of the inner function relates to the backward shift's approximate point spectrum.

Abstract

By a famous result, functions in backward shift invariant subspaces in Hardy spaces are characterized by the fact that they admit a pseudocontinuation a.e. on $\T$. More can be said if the spectrum of the associated inner function has holes on $\T$. Then the functions of the invariant subspaces even extend analytically through these holes. We will discuss the situation in weighted backward shift invariant subspaces. The results on analytic continuation will be applied to consider some embeddings of weighted invariant subspaces into their unweighted companions. Such weighted versions of invariant subspaces appear naturally in the context of Toeplitz operators. A connection between the spectrum of the inner function and the approximate point spectrum of the backward shift in the weighted situation is established in the spirit of results by Aleman, Richter and Ross.