Some generalizations of numerical radius on off-diagonal part of $2\times 2$ operator matrices (1706.05040v1)

Abstract: We generalize several inequalities involving powers of the numerical radius for off-diagonal part of $2\times2$ operator matrices of the form $T=\left[\begin{array}{cc} 0&B, C&0 \end{array}\right]$, where $B, C$ are two operators. In particular, if $T=\left[\begin{array}{cc} 0&B, C&0 \end{array}\right]$, then we get \begin{align*} {1\over 2^{{3\over2}(r-1)}}\max{ | \mu |, | \eta | } \leq w^{r}(T)\leq \frac{1}{2^{r+1}} \max{ | \mu |, | \eta | }, \end{align*} where $r\geq 2$ and $ \mu=|(C-B^{})+i(C+B^{})|^{r}+|(B^{}-C)+i(C+B^{})|^{r}$, $ \eta=|(B-C^{})+i(B+C^{})|^{r}+|(C^{}-B)+i(B+C^{})|^{r}$.