Norm inequalities for sums of two basic elementary operators. (English) Zbl 1140.47023

Boumazgour, Mohamed

J. Math. Anal. Appl. 342, No. 1, 386-393 (2008).

A linear mapping \(T\) on a Banach algebra of the form \(T(x)=\sum_{i=1}^na_ixb_i\), where \(a_i, b_i\,(1\leq i\leq n)\) are given, is called an elementary operator. The operator \(M_{ab}(x)=axb\) is called the basic elementary operator. In this paper, the author presents some estimates for the norm of the operator \(M_{A,B}+M_{C,D}\), where \(A, B, C\) and \(D\) are bounded linear operators acting on a complex Hilbert space \(H\). He also gives necessary and sufficient conditions on nonzero operators \(A, B, C\) and \(D\) such that \(\| M_{A,B}+M_{C,D}\| \) is equal to its optimal value \(\| A\|\| B\| +\| C\| \| D\| \).
For further results in this direction, see also R. M. Timoney [J. Oper. Theory 57, No. 1, 121–145 (2007; Zbl 1150.47025)].

Reviewer: Mohammad Sal Moslehian (Mashhad)

MSC:

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

Keywords:

norm; elementary operator; numerical range; \(C^*\)-algebra

Citations:

Zbl 1150.47025

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