Weighted Bergman spaces: shift-invariant subspaces and input/state/output linear systems

TL;DR

This paper explores the structure of shift-invariant subspaces in weighted Bergman spaces, relating them to linear systems and operator models, extending classical Hardy space results.

Contribution

It generalizes the characterization of shift-invariant subspaces from Hardy spaces to weighted Bergman spaces, connecting them with linear system theory and operator models.

Findings

Shift-invariant subspaces in weighted Bergman spaces can be characterized similarly to Hardy spaces.

Connections between linear systems and subspace structures are established in the weighted Bergman context.

Extensions of classical results to weighted Bergman spaces are demonstrated.

Abstract

It is well known that subspaces of the Hardy space over the unit disk which are invariant under the backward shift occur as the image of an observability operator associated with a discrete-time linear system with stable state-dynamics, as well as the functional-model space for a Hilbert space contraction operator, while forward shift-invariant subspaces have a representation in terms of an inner function. We discuss several variants of these statements in the context of weighted Bergman spaces on the unit disk.