New upper bounds for the numerical radius of Hilbert space operators (Q1997745)

Several estimates are obtained for the numerical radius \(w\) of various combinations of bounded linear operators acting on a Hilbert space. A typical example: \[ w^{2r}\left(\sum_{i=1}^nA_i\right)\leq \frac{n^{2r-1}}{4}\left|\left|\sum_{i=1}^n\left(\left|A_i\right|^{2r}+(\left|A_i^*\right|^{2r}\right) \right|\right|+\frac{n^{2r-1}}{2} \left(\sum_{i=1}^n w\left(\left|A_i\right|^{r}\left|A_i^*\right|^{r} \right)\right) \] for all \(r\geq 1\). Here \(|X|\) stands for the positive factor in the polar decomposition of \(X\): \(|X|=(X^*X)^{1/2}\).