Closed-range posinormal operators and their products

TL;DR

This paper investigates conditions under which the product of two posinormal operators remains posinormal, focusing on closed-range properties, commuting operators, and connections to EP operators, including a new proof of a key theorem.

Contribution

It provides new necessary and sufficient conditions for posinormal operators to have closed range and for their products to be posinormal, extending existing theory.

Findings

Identified conditions for closed-range posinormal operators.

Established criteria for the product of commuting posinormal operators to be posinormal.

Provided a new proof of the Hartwig-Katz Theorem for finite-dimensional spaces.

Abstract

We focus on two problems relating to the question of when the product of two posinormal operators is posinormal, giving (1) necessary conditions and sufficient conditions for posinormal operators to have closed range, and (2) sufficient conditions for the product of commuting closed-range posinormal operators to be posinormal with closed range. We also discuss the relationship between posinormal operators and EP operators (as well as hypo-EP operators), concluding with a new proof of the Hartwig-Katz Theorem, which characterizes when the product of posinormal operators on $\CC^n$ is posinormal.