Bulletin of the Belgian Mathematical Society - Simon Stevin

A note on the norm of a basic elementary operator

Mohamed Boumazgour and Mohamed Barraa

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Abstract

Let ${\cal L}(E)$ be the algebra of all bounded linear operators on a Banach space $E$. For $A,B\in{\cal L}(E)$, define the basic elementary operator $M_{A,B}$ by $M_{A,B}(X)=AXB$, ($X\in{\cal L}(E)$). If $\cal S$ is a symmetric norm ideal of ${\cal L}(E)$, we denote $M_{{\cal S},A,B}$ the restriction of $M_{A,B}$ to $\cal S$. In this paper, the norm equality $\|I+M_{{\cal S},A,B}\|=1+\|A\|\|B\|$ is studied. In particular, we give necessary and sufficient conditions on $A$ and $B$ for this equality to hold in the special case when $E$ is a Hilbert space and $\cal S$ is a Schatten $p$-ideal of ${\cal L}(E)$.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 22, Number 4 (2015), 603-610.

Dates
First available in Project Euclid: 18 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1447856062

Digital Object Identifier
doi:10.36045/bbms/1447856062

Mathematical Reviews number (MathSciNet)
MR3429174

Zentralblatt MATH identifier
1348.47028

Subjects
Primary: 47A12: Numerical range, numerical radius 47A30: Norms (inequalities, more than one norm, etc.) 47B47: Commutators, derivations, elementary operators, etc.

Keywords
Norms elementary operators norm ideals numerical range

Citation

Boumazgour, Mohamed; Barraa, Mohamed. A note on the norm of a basic elementary operator. Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 4, 603--610. doi:10.36045/bbms/1447856062. https://projecteuclid.org/euclid.bbms/1447856062


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