The applications of Cauchy-Schwartz inequality for Hilbert modules to elementary operators and i.p.t.i. transformers

TL;DR

This paper uses the Cauchy-Schwartz inequality in Hilbert modules to provide simple proofs and generalizations of various operator inequalities related to elementary operators and inner product integral transformers.

Contribution

It introduces a unified approach leveraging the Cauchy-Schwartz inequality to simplify and extend existing operator inequalities in Hilbert modules.

Findings

Simplified proofs of existing operator inequalities

Generalization of inequalities for elementary operators

Extension to inner product integral transformers

Abstract

We apply the inequality $\left|\left<x,y\right>\right|\le\|x\|\,\left<y,y\right>^{1/2}$ to give an easy and elementary proof of many operator inequalities for elementary operators and inner type product integral transformers obtained during last two decades, which also generalizes all of them.