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.
Categories: math.FA math.OA