Operators with compatible ranges

TL;DR

This paper introduces the class of compatible range (CoR) operators, extending the concept of disjoint range (DR) operators, and provides new theoretical results on their properties and relationships.

Contribution

The paper defines the broader class of CoR operators, extending the existing DR operator framework, and offers new theoretical insights and results about their structure and behavior.

Findings

Introduced the class of CoR operators as a generalization of DR operators.

Extended and improved results related to DR operators within the CoR framework.

Derived new theoretical properties of CoR operators.

Abstract

A bounded operator $T$ on a finite or infinite--dimensional Hilbert space is called a disjoint range (DR) operator if $R(T)\cap R(T^*)=\{0\}$, where $T^*$ stands for the adjoint of $T$, while $R(\cdot)$ denotes the range of an operator. Such operators (matrices) were introduced and systematically studied by Baksalary and Trenkler, and later by Deng et al. In this paper we introduce a wider class of operators: we say that $T$ is a compatible range (CoR) operator if $T$ and $T^*$ coincide on $R(T)\cap R(T^*)$. We extend and improve some results about DR operators and derive some new results regarding the CoR class.