## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

### A note on the norm of a basic elementary operator

Mohamed Boumazgour and Mohamed Barraa

#### 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