Generalized Cauchy–Schwarz Inequalities and <i>A</i>-Numerical Radius Applications

Abstract

The purpose of this research paper is to introduce new Cauchy–Schwarz inequalities that are valid in semi-Hilbert spaces, which are generalizations of Hilbert spaces. We demonstrate how these new inequalities can be employed to derive novel A-numerical radius inequalities, where A denotes a positive semidefinite operator in a complex Hilbert space. Some of our novel A-numerical radius inequalities expand upon the existing literature on numerical radius inequalities with Hilbert space operators, which are important tools in functional analysis. We use techniques from semi-Hilbert space theory to prove our results and highlight some applications of our findings.

Keywords

• Cauchy–Schwarz inequality
• semi-Hilbert space
• positive operator
• A-numerical radius
• operator A-seminorm