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Computation of antieigenvalues of bounded linear operators via centre of mass

Kallol Paul, Gopal Das, Lokenath Debnath

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 $θ$-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 $θ$-antieigenvalue for an operator acting on a finite dimensional Hilbert space.

Computation of antieigenvalues of bounded linear operators via centre of mass

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 -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 -antieigenvalue for an operator acting on a finite dimensional Hilbert space.

Paper Structure

This paper contains 3 sections, 7 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. $\theta$-antieigenvalues
  3. Total antieigenvalue and $\theta$-antieigenvalue

Key Result

Theorem 1

Let f be a unit $\theta$-antieigenvector of a bounded linear operator T. Then f satisfies the following equation where $A = Re T$, $B = Im T$, $a = Re \langle Tf,f \rangle$, $b = Im \langle Tf,f \rangle$.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2: $\theta$-antieigenvalue
  • Definition 3
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Theorem 4
  • Lemma 2
  • ...and 1 more