Partial orders on partial isometries

TL;DR

This paper explores three natural pre-orders on non-unitary partial isometries with equal defect indices, linking their comparison to bounded or isometric multipliers between associated reproducing kernel Hilbert spaces, and analyzing their properties.

Contribution

It introduces a novel characterization of pre-order relations on partial isometries via multipliers between specific Hilbert spaces, connecting operator theory with function space analysis.

Findings

Pre-orders are characterized by the existence of bounded or isometric multipliers.

Model subspaces and deBranges-Rovnyak spaces are used to realize these Hilbert spaces.

The paper investigates properties and equivalence classes generated by these pre-orders.

Abstract

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries these spaces can be realized as the well-known model subspaces and deBranges-Rovnyak spaces. This characterization is applied to investigate properties of these pre-orders and the equivalence classes they generate.