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A note on the norm of a basic elementary operator. (English) Zbl 1348.47028
Summary: Let $$\mathcal{L}(E)$$ be the algebra of all bounded linear operators on a Banach space $$E$$. For $$A,B\in\mathcal{L}(E)$$, define the basic elementary operator $$M_{A,B}$$ by $$M_{A,B}(X)=AXB$$, ($$X\in\mathcal{L}(E)$$). If $$\mathcal{S}$$ is a symmetric norm ideal of $$\mathcal{L}(E)$$, we denote $$M_{\mathcal{S},A,B}$$ the restriction of $$M_{A,B}$$ to $$\mathcal{S}$$. In this paper, the norm equality $$\| I+M_{\mathcal{S},A,B}\|=1+\| A\|\| B\|$$ 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 $$\mathcal{S}$$ is a Schatten $$p$$-ideal of $$\mathcal{L}(E)$$.
##### MSC:
 47B47 Commutators, derivations, elementary operators, etc. 47A12 Numerical range, numerical radius 47A30 Norms (inequalities, more than one norm, etc.) of linear operators
##### Keywords:
norms; elementary operators; norm ideals; numerical range
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