More on $\omega$-orthogonality and $\omega$-parallelism

Abstract

We investigate some aspects of various numerical radius orthogonalities and numerical radius parallelism for bounded linear operators on a Hilbert space $\mathcal{H}$. Among several results, we show that if $T,S\in \mathbb{B}(\mathcal{H})$ and $M_{\omega (T)}^{\ast }=M_{\omega (S)}^{\ast }$, then $T{\perp }_{\omega B}S$ if and only if $S{\perp }_{\omega B}T$, where $M_{\omega (T)}^{\ast }=\{\{x_n\}:\Vert x_n\Vert =1,{\lim }_n|⟨T{x}_n,x_n⟩|=\omega (T)\}$, and $\omega (T)$ is the numerical radius of $T$ and ${\perp }_{\omega B}$ is the numerical radius Birkhoff orthogonality.