Cosine of angle and center of mass of an operator

TL;DR

This paper introduces the concepts of real and total center of mass for bounded linear operators, explores their relation with cosine measures, and provides new proofs and methods for calculating antieigenvalues.

Contribution

It presents a novel framework for understanding operator center of mass and extends the min-max equality using operator orthogonality, with practical calculation methods.

Findings

Established relations between center of mass and cosine of operators

Provided a new proof of the Min-max equality

Offered alternative methods for calculating antieigenvalues

Abstract

We consider the notion of real center of mass and total center of mass of a bounded linear operator relative to another bounded linear operator and explore their relation with cosine and total cosine of a bounded linear operator acting on a complex Hilbert space. We give another proof of the Min-max equality and then generalize it using the notion of orthogonality of bounded linear operators. We also illustrate with examples an alternative method of calculating the antieigenvalues and total antieigenvalues for finite dimensional operators.