Computation of antieigenvalues of bounded linear operators via centre of mass

TL;DR

This paper introduces theta-antieigenvalues for bounded linear operators on complex Hilbert spaces, relating them to the centre of mass and unifying various existing antieigenvalue concepts with computational examples.

Contribution

It defines theta-antieigenvalues, connects them to the centre of mass, and shows they generalize existing antieigenvalue notions, providing a unified framework.

Findings

Theta-antieigenvalues relate to the centre of mass of operators.

Existing antieigenvalues are special cases of theta-antieigenvalues.

Examples demonstrate calculation of theta-antieigenvalues on finite-dimensional spaces.

Abstract

We introduce the concept of theta-antieigenvalue and theta-antieigenvector of a bounded linear operator on complex Hilbert space. We study the relation between theta-antieigenvalue and centre of mass of a bounded linear operator and compute antieigenvalue using the relation. This follows the notion of symmetric antieigenvalues introduced by Hossein et al. in \cite{19}. We show that the concept of real antieigenvalue, imaginary antieigenvalue and symmetric antieigenvalue follows as a special case of theta-antieigenvalue. We also show how the concept of total antieigenvalue is related to the $\theta$-antieigenvalue. In fact, we show that all the concepts of antieigenvalues studied so far follows from the concept of theta-antieigenvalue. We illustrate with example how to calculate the $\theta$-antieigenvalue for an operator acting on a finite dimensional Hilbert space.