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

Abstract

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