Sharp inequalities for the numerical radius of block operator matrices

Abstract

In this paper, we present several sharp upper bounds for the numerical radii of the diagonal and off-diagonal parts of the $2\times 2$ block operator matrix $\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]$. Among extensions of some results of Kittaneh et al., it is shown that if $T=\left[\begin{array}{cc}A& 0\\ 0& D\end{array}\right]$, and $f$ and $g$ are non-negative continuous functions on $[0,\infty )$ such that $f(t)g(t)=t(t\ge 0)$, then for all nonnegative nondecreasing convex functions $h$ on $[0,\infty )$ , we obtain that \begin{align*}h\left(w^r(T)\right)\leq \max\left(\left\|\frac{1}{p}h\left(f^{pr}(\left|A\right|)\right)+ \frac{1}{q}h\left(g^{qr}(\left|A^*\right|)\right)\right\|, \left\|\frac{1}{p}h\left(f^{pr}(\left|D\right|)\right)+ \frac{1}{q}h\left(g^{qr}(\left|D^*\right|)\right)\right\|\right), \end{align*} where $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$ and $r\min (p,q)\ge 2$.

18 pages, to appear in Analysis Mathematica

References in corpus (2)

• Upper bounds for numerical radius inequalities involving off-diagonal operator matrices
• Some generalizations of the Aluthge transform of operators