Hypo-EP Matrices of Adjointable Operators on Hilbert    C  ∗  -Modules

Xiaopeng Li,Junjie Huang,Alatancang Chen

doi:10.1155/2021/5528089

Abstract

This paper introduces and studies hypo-EP matrices of adjointable operators on Hilbert C ∗ -modules, based on the generalized Schur complement. The necessary and sufficient conditions for some modular operator matrices to be hypo-EP are given, and some special circumstances are also analyzed. Furthermore, an application of the EP operator in operator equations is given.

Highlights

Introduction and Preliminaries e EP matrix, as an extension of the normal matrix, was proposed by Schwerdtfeger; a square matrix T over the complex field C is said to be an EP matrix if T and T∗ share the same range [1, 2]. e notion of EP matrices was extended by Campbell and Meyer to operators with closed range on a Hilbert space in [3]
It is well known that R(T) is closed if and only if the Moore–Penrose inverse T† of T exists and that T is an EP operator if and only if T†T TT†
It is shown that T is a hypo-EP operator if and only if T†T2T† TT†. e hypo-EP operator is our focus of attention in this paper, and it has been studied in [10, 11]. e EP operator can be applied to the solution of operator equations, see Section 3 of this article. e properties of hypo-EP and EP operators can find applications in reverse order law [12] and core partial order [13] and will be useful in some other applied fields [14, 15]

Summary

Introduction

Introduction and Preliminaries eEP matrix, as an extension of the normal matrix, was proposed by Schwerdtfeger; a square matrix T over the complex field C is said to be an EP matrix if T and T∗ share the same range [1, 2]. e notion of EP matrices was extended by Campbell and Meyer to operators with closed range on a Hilbert space in [3]. Is paper introduces and studies hypo-EP matrices of adjointable operators on Hilbert C∗-modules, based on the generalized Schur complement. The range of an EP or a hypo-EP operator on Hilbert C∗-modules is not necessarily closed, and we further have the following properties. Let H be a Hilbert A-module and T ∈ L(H) with closed range. Using generalized Schur complements, we study the hypo-EP property of matrices of adjointable operators on Hilbert C∗-modules.

Results

Conclusion