Some numerical radius inequalities for Hilbert space operators
Let \(B(H)\) denote the \(C^*\) algebra of all bounded operators on a complex Hilbert space \(H\) with inner product \(\langle\cdot,\cdot\rangle\). The authors treat numerical radius and related inequalities in the present paper. Their main result says that, if \(A,B,C,D \in B(H)\) and \(T = \left[\begin{smallmatrix} A &B\\ C &D \end{smallmatrix}\right]\), then \(\max(w(A),w(D))\leq\frac12(\| T\| + \| T^2\|^{\frac12})\) and \(\max((w(BC)^{\frac12}, w(CB)^{\frac12})\leq\frac12(\|T\|+\|T^2\|^{\frac12})\). Moreover, they prove that, if \(A \in B(H)\) is positive, then \(w(AX-XA)\leq\frac12\|A\|(\|X\|+\|X^2\|^{\frac12})\). The authors both sharpen and provide some new proofs of previously established inequalities and they also mention some applications.
scientific article; zbMATH DE number 5673995