Closed $ EP $ and Hypo-$ EP $ Operators on Hilbert Spaces

TL;DR

This paper extends the concepts of EP and hypo-EP operators from bounded to densely defined closed operators on Hilbert spaces, exploring their properties and relationships in the unbounded operator context.

Contribution

It introduces definitions and extends existing results for EP and hypo-EP operators to unbounded closed operators on Hilbert spaces.

Findings

Extended EP and hypo-EP operator definitions to unbounded operators.

Established properties and relationships for these operators in the unbounded setting.

Provided theoretical framework for future research on unbounded operator classes.

Abstract

A bounded linear operator $ A$ on a Hilbert space $ \mathcal H $ is said to be an $ EP $ (hypo-$ EP $) operator if ranges of $ A $ and $ A^* $ are equal (range of $ A $ is contained in range of $ A^* $) and $ A $ has a closed range. In this paper, we define $EP$ and hypo-$EP$ operators for densely defined closed linear operators on Hilbert spaces and extend results from bounded operator settings to (possibly unbounded) closed operator settings.