IJPAM: Volume 118, No. 3 (2018)

Title

A NOTE ON THE JOINT ESSENTIAL NUMERICAL
RANGE OF ALUTHGE TRANSFORM

Authors

O.S. Cyprian
Department of Mathematics
Kibabii University
P.O. Box 1699-50200, Bungoma, KENYA

Abstract

Denote by $T$ the bounded linear operator on a complex Hilbert space $X$ and let $T = U \vert T\vert$ be any polar decomposition of $T$ with $U$ a partial isometry and $\vert T\vert = (T^{*}T)^{\frac{1}{2}}$. Then, the Aluthge transform $\widetilde{T}$ of $T$ is the operator $\vert T\vert^{\frac{1}{2}} U \vert T\vert^{\frac{1}{2}}$. This study of the Aluthge transform $\widetilde{T}$ was introduced and studied by Aluthge in his study of $p$-hyponormal operators in 1990. This notion has received much attention in recent years for a single operator $T$. For instance, quite a lot has been researched on the essential numerical range of $\widetilde{T}$ of an operator $T.$ In contrast to this, nothing is known about the joint essential numerical range of Aluthge transform $\widetilde{T}$ of an $m-$tuple operator $T = (T_{1},...,T_{m}).$ The focus of this note is to study the properties of the joint essential numerical range of Aluthge transform for an $m-$tuple operator $T = (T_{1},...,T_{m}).$ This study is therefore helpful in the development of the research on hyponormal operators and semi-hyponormal operators.

History

Received: July 11, 2017
Revised: March 6, 2018
Published: April 11, 2018

AMS Classification, Key Words

AMS Subject Classification: 47LXX
Key Words and Phrases: Aluthge transform, essential numerical range, joint essential numerical range, Hilbert space

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Bibliography

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How to Cite?

DOI: 10.12732/ijpam.v118i3.7 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 118
Issue: 3
Pages: 573 - 579


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