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

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)].

### 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

Zbl 1150.47025
Full Text:

### References:

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