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 $\mathscr{H}$. Among several results, we show that if $T,S\in \mathbb{B}(\mathscr{H})$ and $M^*_{\omega(T)}=M^*_{\omega(S)}$, then $T\perp_{\omega B} S$ if and only if $S\perp_{\omega B} T$, where $M^*_{\omega(T)}=\{\{x_n\}:\,\,\,\|x_n\|=1, \lim_n|\langle Tx_n, x_n\rangle|=\omega(T)\}$, and $\omega(T)$ is the numerical radius of $T$ and $\perp_{\omega B}$ is the numerical radius Birkhoff orthogonality.