Numerical radius inequalities for products and commutators of operators (Q2786836)

The authors present several numerical radius inequalities involving commutators and products of operators belonging to the \(C^\ast\)-algebra \(B(H)\) of all bounded linear operators acting on a Hilbert space \(H\). Their work is based on the known inequality \(w(A)=\sup_{\theta\in\mathbb{R}}\|\operatorname {Re}(e^{i\theta}A)\|\), where \(w(A)\) denotes the numerical radius of \(A\in B(H)\); see also [the second author et al., Linear Algebra Appl. 471, 46--53 (2015; Zbl 1322.47008)]. They prove that, for \(A,B,X,Y\in B(H)\), the following inequlity holds NEWLINE\[NEWLINEw(A^\ast XB^\ast\pm BYA)\leq \left\|\frac{\| B\|}{\| A\|}|A|^2+\frac{\| A\|}{\| B\|}|B^\ast|^2\right\| w\left( \left[\begin{matrix} 0 & X \\ Y & 0 \end{matrix}\right]\right).NEWLINE\]NEWLINE This improves NEWLINE\[NEWLINEw(A^\ast XB^\ast\pm BYA)\leq 2\| A\|\,\| B\| w\left( \left[\begin{matrix} 0 & X \\ Y & 0 \end{matrix}\right]\right)NEWLINE\]NEWLINE obtained in [\textit{O. Hirzallah} et al., Numer. Funct. Anal. Optim. 32, No. 7, 739--749 (2011; Zbl 1227.47003)]. They also extend the inequality NEWLINE\[NEWLINEw( AX\pm XA)\leq 2\sqrt{2} \| A\| w(X)NEWLINE\]NEWLINE of \textit{C.-K. Fong} and \textit{J.A.R. Holbrook} [Can. J. Math. 35, 274--299 (1983; Zbl 0477.47005)] to NEWLINE\[NEWLINEw(AX\pm XA)\leq \sqrt{\|\, |A|^2+|A^\ast|^2\|}\sqrt{\|\, |X|^2+|X^\ast|^2\|},NEWLINE\]NEWLINE where \(A, X \in B(H)\). Finally, they show that NEWLINE\[NEWLINEw(AB) \leq \frac{1}{2}w(BA)+\frac{1}{4}\left\|\,\frac{\| B\|}{\| A\|}|A|^2+\frac{\| A\|}{\| B\|}|B^\ast|^2\right\|NEWLINE\]NEWLINE for all \(A, B \in B(H)\), which gives rise to some inequalities of the authors [J. Oper. Theory 72, No. 2, 521--527 (2014; Zbl 1389.47014)].