On a preorder relation for contractions

TL;DR

This paper introduces a preorder relation for contractions on Hilbert spaces based on invariant subspace restrictions, explores its equivalence classes, and extends previous results on nonunitary partial isometries.

Contribution

It defines a new preorder relation for contractions, analyzes the structure of its equivalence classes, and generalizes known results for nonunitary partial isometries.

Findings

Identifies conditions where equivalence classes coincide with unitary equivalence classes.

Extends previous work by Garcia, Martin, and Ross on nonunitary partial isometries.

Provides a framework for understanding contractions via invariant subspace restrictions.

Abstract

An order relation for contractions on a Hilbert space can be introduced by stating that $A\preccurlyeq B$ if and only $A$ is unitarily equivalent to the restriction of $B$ to an invariant subspace. We discuss the equivalence classes associated to this relation, and identify cases in which they coincide with classes of unitary equivalence. The results extend those for completely nonunitary partial isometries obtained by Garcia, Martin, and Ross.