Remarks on Essential Maximal Numerical Range of Aluthge Transform

Omukhwaya Sakwa Cyprian,Shem Aywa,Lucy Chikamai

doi:10.18052/www.scipress.com/ijpms.20.1

Abstract

This paper focuses on the properties of the essential maximal numerical range of Aluthgetransform T. For instance, among other results, we show that the essential maximal numerical rangeof Aluthge transform is nonempty and convex. Further, we prove that the essential maximal numericalrange of Aluthge transform e T is contained in the essential maximal numerical range of T. This studyis therefore an extention of the research on Aluthge transform which was begun by Aluthge in hisstudy of p−hyponormal operators.

Highlights

Let B(X) denote the algebra of bounded linear operators acting on a complex Hilbert space X
We denote by the bounded linear operator on a complex Hilbert space X and let T = U |T | be any polar decomposition of T
This paper established some of the properties of the essential maximal numerical range of Aluthge transform T

Summary

Introduction

Let B(X) denote the algebra of bounded linear operators acting on a complex Hilbert space X. MaxWe(T ) = {r ∈ C : ⟨T xn, xn⟩ → r, xn → 0 weakly and ∥T xn∥ → ∥T ∥e}. Assume without loss of generality that for a point r ∈ C there exists an orthonormal sequence {xn} ∈ X such that ⟨T xn, xn⟩ → r and ∥T xn∥ → ∥T ∥e.

Results

Conclusion