A note on unitary equivalence of operators acting on reproducing kernel Hilbert spaces

TL;DR

This paper presents a new criterion for determining when operators on reproducing kernel Hilbert spaces are unitarily equivalent, extending previous theorems and analyzing the structure of intertwining operators and Cowen-Douglas operators.

Contribution

It introduces an alternative, equivalent criterion for unitary equivalence of operators on reproducing kernel Hilbert spaces, and explores the uniqueness of Cowen-Douglas operator decompositions.

Findings

Established a new criterion for unitary equivalence.

Described the structure of intertwining operators.

Proved the uniqueness of Cowen-Douglas operator decomposition.

Abstract

A well-known theorem due to R. E. Curto and N. Salinas gives a necessary and sufficient condition for the unitary equivalence of commuting tuples of bounded linear operators acting on reproducing kernel Hilbert spaces. Inspired by this theorem, we obtain a different but equivalent criterion for the unitary equivalence of operators acting on reproducing kernel Hilbert spaces. As an application, we describe the structure of intertwining operator and prove that the decomposition of Cowen-Douglas operators is unique up to unitary equivalence.