SOME GENERALIZTIONS OF NUMERICAL RADIUS INEQUALITIES INCLUDING THE OFF-DIAGONAL PARTS OF BLOCK MATRICES

Abstract

\begin{abstract}
In this paper, we introduce several numerical radius inequalities including
off-diagonal part of $2\times 2$ positive semidefinite block matrices and
their diagonal blocks. It is shown that if $A,B,C\in {\mathbb{M}}}_{n}$ are such that

$\left[
\begin{array}{cc}
A & B^{\ast } \\
B & C%
\end{array}%
\right] \geq 0$,

then

\begin{equation*}
w^{2r}(B)\leq \frac{1}{2}\sqrt{\left\Vert A^{4r\alpha }+A^{4r(1-\alpha
)}\right\Vert \left\Vert C^{4r\alpha }+C^{4r(1-\alpha )}\right\Vert }
\end{equation*}

and

\begin{equation*}
w^{2r}(B)\leq \left\Vert \alpha A^{\frac{r}{\alpha }}+(1-\alpha )C^{\frac{r}{%
1-\alpha }}\right\Vert ,
\end{equation*}

for $0<\alpha <1$, $r\geq 1.$

Moreover, we establish some numerical radius inequalities for products and
sums of matrices.