Conditions Implying Convexoidity and Normaloidity

International Journal of Research and Innovation in Applied Science (IJRIAS) | Volume V, Issue X, October 2020 | ISSN 2454–6186

 Conditions Implying Convexoidity and Normaloidity

G.M. Kariuki, M. Kavila
Department of Mathematics and Actuarial Science Kenyatta University P.O.Box 43844-00100, Nairobi, Kenya

Abstract: Let B(H) denote the algebra of bounded linear operators on a Hilbert space H into itself. Our task in this note is to prove conditions that imply convexoidity and normaloidity. It is shown among other results that if T is normaloid then Tk is normaloid for k ∈ N.
AMS Subject classification: 47B47, 47A30, 47B20

Key words: normal, hyponormal, convexoid and normaloid operators.

I. INTRODUCTION

Let B(H) denote the algebra of bounded operators on a complex Hilbert space H into itself. Istrăţescu [3] proved the following result.
Theorem A. Let T ∈ B(H) be a hyponormal operator. Then T − µ is also hyponormal for any µ ∈ C.
Sheth [4] also proved the following result involving self-adjoint operators.
Theorem B. Let T ∈ B(H). Then T is normaloid if it is self-adjoint.
In this paper we will outline an alternative proof to Theorem B and also give other conditions that imply normaloidity and convexoidity.
The following lemma by Blumenson [1] will also act as a stepping stone to the main theorem in this paper involving normaloid operators.
Lemma C. Let T ∈ B(H). Then
1/2||T||≤w(T)≤ ‖T‖
r(T)≤w(T)≤ ‖T‖

II. NOTATION AND TERMINOLOGY

Given an operator T ∈ B (H), we shall denote the spectral radius and the numerical radius of T by r(T) and w(T) respectively.
r(T) = Sup {|λ|: λ ∈ σ(T)}
w (T) = Sup {|λ|: λ ∈ W (T)} where W(T) and σ(T) denote the numerical range of T and the spectrum of T respectively given by:
W (T) = {〈Tx,x.〉: ||x|| = 1, x ∈ H}
σ (T) = {λ ∈ C: T − λI is not invertible}
An operator T is said to be:
Self-adjoint if T = T^*

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