Some numerical radius inequalities for Hilbert space operators

Abstract

We present several numerical radius inequalities for Hilbert space operators. More precisely, we prove that if $A,B,C,D\in B\left(H\right)$ and $T=\left[\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill C\hfill & \hfill D\hfill \end{array}\right]$ then $max\left(w\left(A\right),w\left(D\right)\right)\le \left(1∕2\right)\left(\parallel T\parallel +\parallel T^2{\parallel }^{1∕2}\right)$ and $max\left({\left(w\left(BC\right)\right)}^{1∕2},{\left(w\left(CB\right)\right)}^{1∕2}\right)\le \left(1∕2\right)\left(\parallel T\parallel +\parallel T^2{\parallel }^{1∕2}\right)$. We also show that if $A\in B\left(H\right)$ is positive, then

$$w\left(AX-XA\right)\le \frac{1}{2}\parallel A\parallel \left(\parallel X\parallel +\parallel X^2{\parallel }^{1∕2}\right).$$