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Numerical radius inequalities for commutators of Hilbert space operators. (English) Zbl 1227.47003
For a bounded linear operator $$T$$ on a complex Hilbert space $$H$$, the numerical range is given by $$W(T)=\{ \langle Tx,x\rangle:~x\in H, ~\| x\|=1\}$$. The quantity $$w(T)=\sup\{ |\lambda|:\lambda \in W(T)\}$$ is the numerical radius of $$T$$. It is well known that $$w(\cdot)$$ is a norm on $$B(H)$$, the algebra of all bounded linear operators on $$H$$, which is equivalent to the operator norm. More precisely, $$\tfrac{1}{2}\| T\| \leq w(T)\leq \|T\|$$. The numerical radius is not submultiplicative, moreover, even the inequality $$w(ST)\leq \| S\| w(T)$$ does not hold in general. However, there is an interesting inequality which does hold true. Namely, for arbitrary operators $$S, T\in B(H)$$, one has $$w(ST+TS^*)\leq 2\| S\| w(T)$$. In the paper under review, some natural generalizations of this last and some other inequalities are obtained. For instance, let $$A, B, S, T\in B(H)$$ be arbitrary, then $$w(ASB^*\pm BTA^*)\leq 2\| A\| \| B\| w(X),$$ where $$X\in B(H\oplus H)$$ is an operator with the block matrix which has $$0$$ on the diagonal and operators $$S$$ and $$T$$ off the diagonal.

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