Cartesian decomposition and Numerical radius inequalities

Abstract

We show that if $T=H+iK$ is the Cartesian decomposition of $T\in \mathbb{B}(\mathcal{H})$, 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 $A,B,X\in \mathbb{B}(\mathcal{H})$ and $0\le mI\le X$, 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_{θ\in \mathbb{R}}\left\Vert (AX-XB)+e^{iθ}(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.

8 pages, appeared in Linear Algebra Appl

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