Cartesian decomposition and Numerical radius inequalities

Abstract: We show that if $T=H+iK$ is the Cartesian decomposition of $T\in \mathbb{B(\mathscr{H})}$, then for $\alpha ,\beta \in \mathbb{R}$, $\sup_{\alpha ^{2}+\beta ^{2}=1}\Vert \alpha H+\beta K\Vert =w(T)$. We then apply it to prove that if $A,B,X\in \mathbb{B(\mathscr{H})}$ and $0\leq mI\leq X$, then \begin{align*} m\Vert \mbox{Re}(A)-\mbox{Re}(B)\Vert & \leq w(\mbox{Re}(A)X-X\mbox{Re}(B)) \ & \leq \frac{1}{2}\sup_{\theta \in \mathbb{R}}\left\Vert (AX-XB)+e^{i\theta }(XA-BX)\right\Vert \ & \leq \frac{\Vert AX-XB\Vert +\Vert XA-BX\Vert }{2}, \end{align*} where $\mbox{Re}(T)$ denotes the real part of an operator $T$. A refinement of the triangle inequality is also shown.