The $L^2$-Cartesian Numerical Radius of Hilbert Space Operators

Let $\mathcal{H}$ be a complex Hilbert space, and let $T=H+iK\in \mathcal{B}(\mathcal{H})$ be the Cartesian decomposition of $T$, where $H=H^{\ast }$ and $K=K^{\ast }$. We introduce the $L^2$-Cartesian numerical radius $${\omega }_2(T)={(\frac{1}{2\pi }{\int }_0^{2\pi }\mathrm{\parallel }\cos \theta H+\sin \theta K{\mathrm{\parallel }}^{2}d\theta )}^{1\mathrm{/}2}\mathrm{.}$$ We study its basic properties and compare it with the classical numerical radius $\omega (T)$. In particular, we establish the sharp estimate $$\frac{1}{\sqrt{2}}\omega (T)\le {\omega }_2(T)\le \omega (T),$$ and establish lower and upper bounds in terms of the Cartesian parts $H,K$. As a product, we obtain $$\frac{1}{4}\parallel T^{\ast }T+T{T}^{\ast }\parallel \le {\omega }_2(T{)}^2\le \omega (T{)}^2\mathrm{.}$$