## Annals of Functional Analysis

### A note concerning the numerical range of a basic elementary operator

#### Abstract

Let $\mathcal{B}(H)$ be the algebra of all bounded linear operators on a complex Hilbert space $H$, and let $\mathcal{S}$ be a norm ideal in $\mathcal{B}(H)$. For $A,B\in\mathcal{B}(H)$, define the elementary operator $M_{\mathcal{S},A,B}$ on $\mathcal{S}$ by $M_{\mathcal{S},A,B}(X)=AXB$ ($X\in\mathcal{S}$). The aim of this paper is to give necessary and sufficient conditions under which the equality $V(M_{\mathcal{S},A,B})=\overline{\operatorname{co}}(W(A)W(B))$ holds. Here $V(T)$ and $W(T)$ denote the algebraic numerical range and spatial numerical range of an operator $T$, respectively, and $\overline{\operatorname{co}}(\Omega)$ denotes the closed convex hull of a subset $\Omega\subseteq{\mathbb{C}}$.

#### Article information

Source
Ann. Funct. Anal., Volume 7, Number 3 (2016), 434-441.

Dates
Accepted: 14 January 2016
First available in Project Euclid: 17 June 2016

https://projecteuclid.org/euclid.afa/1466172137

Digital Object Identifier
doi:10.1215/20088752-3605510

Mathematical Reviews number (MathSciNet)
MR3513127

Zentralblatt MATH identifier
1351.47003

#### Citation

Boumazgour, Mohamed; Nabwey, Hossam A. A note concerning the numerical range of a basic elementary operator. Ann. Funct. Anal. 7 (2016), no. 3, 434--441. doi:10.1215/20088752-3605510. https://projecteuclid.org/euclid.afa/1466172137

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