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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
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Full Text: Euclid