Diagonalization of Shift-Preserving Operators

TL;DR

This paper introduces a new concept of s-diagonalization for shift-preserving operators on finitely generated shift-invariant spaces, providing conditions for diagonalizability and a generalized spectral theorem.

Contribution

It defines s-diagonalization, establishes necessary and sufficient conditions for it, and extends the spectral theorem to normal shift-preserving operators.

Findings

Introduced s-diagonalization for shift-preserving operators

Provided criteria for s-diagonalizability based on range operators

Extended spectral theorem to normal shift-preserving operators

Abstract

In this note we study the structure of shift-preserving operators acting on a finitely generated shift-invariant space. We define a new notion of diagonalization for these operators, which we call s-diagonalization. We give necessary and sufficient conditions on a bounded shift-preserving operator in order to be s-diagonalizable. These conditions are in terms of its range operator. We also obtain a generalized Spectral Theorem for normal bounded shift-preserving operators.