The applications of Cauchy-Schwartz inequality for Hilbert modules to elementary operators and I.P.T.I. transformers

Dragoljub Keckic

doi:10.2298/aadm180123002k

Abstract

We apply the inequality |?x,y?| ? ||x|| ?y,y?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 many of them.

Highlights

Let A be a Banach algebra, and let aj, bj ∈ A
We shall deal with unitarily invariant norms on the algebra B(H) of all bounded Hilbert space operators
E is central and x, y ∈ M0 are normal with respect to Φ and some conjugation in any unitarily invariant norm

Summary

INTRODUCTION

Let A be a Banach algebra, and let aj, bj ∈ A. Elementary operators, introduced by Lummer and Rosenblum in [12] are mappings from A to A of the form n (1). A similar mapping, called inner product type integral transformer Cauchy Schwartz inequality, unitarily invariant norm, elementary operator, inner product type transformers. There were obtained a number of inequalities involving elementary operators on B(H) as well as i.p.t.i. type transformers. The aim of this paper is to give an easy and elementary proof of those proved in [7, 8, 6, 9, 4, 21, 10] and [11] using the Cauchy Schwartz inequality for Hilbert. C∗-modules the inequality stated in the abstract, which generalizes all of them

PRELIMINARIES

CAUCHY-SCHWARTZ INEQUALITIES

GRÜSS TYPE INEQUALITIES

CONCLUDING REMARKS