Unitarily invariant norm inequalities for operators

TL;DR

This paper establishes new inequalities involving unitarily invariant norms for operators on Hilbert spaces, including bounds for sums and products of operators and projections, expanding the theoretical framework of operator inequalities.

Contribution

It introduces novel unitarily invariant norm inequalities for sums and products of Hilbert space operators and projections, generalizing existing results in operator theory.

Findings

Proved a bound for sums of operator products using unitarily invariant norms.

Derived inequalities involving projections and their combinations.

Extended the understanding of operator norm inequalities in Hilbert spaces.

Abstract

We present several operator and norm inequalities for Hilbert space operators. In particular, we prove that if $A_{1},A_{2},...,A_{n}\in {\mathbb B}({\mathscr H})$, then \[|||A_{1}A_{2}^{*}+A_{2}A_{3}^{*}+...+A_{n}A_{1}^{*}|||\leq|||\sum_{i=1}^{n}A_{i}A_{i}^{*}|||,\] for all unitarily invariant norms. We also show that if $A_{1},A_{2},A_{3},A_{4}$ are projections in ${\mathbb B}({\mathscr H})$, then &&|||(\sum_{i=1}^{4}(-1)^{i+1}A_{i})\oplus0\oplus0\oplus0|||&\leq&|||(A_{1}+|A_{3}A_{1}|)\oplus (A_{2}+|A_{4}A_{2}|)\oplus(A_{3}+|A_{1}A_{3}|)\oplus(A_{4}+|A_{2}A_{4}|)||| for any unitarily invariant norm.