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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.

47A12 Numerical range, numerical radius
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47B47 Commutators, derivations, elementary operators, etc.
Full Text: DOI
[1] DOI: 10.1016/j.laa.2008.02.004 · Zbl 1153.47005 · doi:10.1016/j.laa.2008.02.004
[2] Bhatia R., Positive Definite Matrices (2007) · Zbl 1125.15300
[3] DOI: 10.4153/CJM-1983-015-3 · Zbl 0477.47005 · doi:10.4153/CJM-1983-015-3
[4] DOI: 10.1007/978-1-4613-8498-4 · doi:10.1007/978-1-4613-8498-4
[5] A Hilbert Space Problem Book (), 2. ed. (1982)
[6] DOI: 10.4064/sm158-1-2 · Zbl 1113.15302 · doi:10.4064/sm158-1-2
[7] Omidvar M., Involve. J. Math. 2 pp 469– (2009)
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