A Class of Sub-Hardy Hilbert Spaces Associated with Weighted Shifts

TL;DR

This paper investigates a class of sub-Hardy Hilbert spaces where multiplication by z acts less strongly than an isometry, characterizing these spaces via weighted shifts and deriving classical results as corollaries.

Contribution

It introduces a new class of sub-Hardy Hilbert spaces linked to weighted shifts, extending classical results of de Branges and Beurling.

Findings

Characterization of sub-Hardy spaces via weighted shifts

Identification of operators weaker than isometries as weighted shifts

Derivation of classical theorems as corollaries

Abstract

In this note we study sub-Hardy Hilbert spaces on which the the action of the operator of multiplication by the coordinate function z is assumed to be weaker than that of an isometry. We identify such operators with a class of weighted shifts. The well known results of de Branges and Beurling are deduced as corollaries .