Weighted Hardy spaces: shift invariant and coinvariant subspaces, linear systems and operator model theory

TL;DR

This paper extends classical operator model theory and shift-invariant subspace analysis from Hardy spaces to weighted Hardy spaces, exploring hypercontraction operators and their associated structures.

Contribution

It develops an analogue of Sz.-Nagy--Foias and Beurling-Lax theories within weighted Hardy spaces and hypercontraction operator frameworks.

Findings

Establishes a correspondence between weighted shift-invariant subspaces and hypercontraction operators.

Generalizes classical model theory to weighted Hardy space context.

Provides new insights into operator structures in weighted function spaces.

Abstract

The Sz.-Nagy--Foias model theory for $C_{\cdot 0}$ contraction operators combined with the Beurling-Lax theorem establishes a correspondence between any two of four kinds of objects: shift-invariant subspaces, operator-valued inner functions, conservative discrete-time input/state/output linear systems, and $C_{\cdot 0}$ Hilbert-space contraction operators. We discuss an analogue of all these ideas in the context of weighted Hardy spaces over the unit disk and an associated class of hypercontraction operators.