The \(L^2\)-Cartesian Numerical Radius of Hilbert Space Operators

Abstract

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 \(H=H^{*}\) and \(K=K^{*}\). We introduce the \(L^2\)-Cartesian numerical radius \[ \omega_2(T)=\left(\frac{1}{2\pi}\int_0^{2\pi}\|\cos\theta H+\sin\theta K\|^2\,d\theta\right)^{1/2}. \] 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 \[ \frac{1}{4}\left\|T^{*}T+TT^{*}\right\|\le \omega_2(T)^2\le \omega(T)^2. \]