New Orders Among Hilbert Space Operators

TL;DR

This paper establishes new order relations among Hilbert space operators, introduces positivity-based techniques for operator inequalities, and extends results on numerical radius and singular values.

Contribution

It presents novel L"{o}ewner orderings among operators and develops positivity methods for deriving operator inequalities in Hilbert spaces.

Findings

New L"{o}ewner partial orderings among operators

Positivity-based techniques for operator inequalities

Extended numerical radius and singular value results

Abstract

This article introduces several new relations among related Hilbert space operators. In particular, we prove some L\"{o}ewner partial orderings among $T, |T|, \mathcal{R}T, \mathcal{I}T, |T|+|T^*|$ and many other related forms, as a new discussion in this field; where $\mathcal{R}T$ and $\mathcal{I}T$ are the real and imaginary parts of the operator $T$. Our approach will be based on proving the positivity of some new matrix operators, where several new forms for positive matrix operators will be presented as a key tool in obtaining the other ordering results. As an application, we present some results treating numerical radius inequalities in a way that extends some known results in this direction, in addition to some results about the singular values.