Numerical radius parallelism of Hilbert space operators

Abstract

In this paper, we introduce a new type of parallelism for bounded linear operators on a Hilbert space $\big(\mathscr{H}, \langle \cdot ,\cdot \rangle\big)$ based on numerical radius. More precisely, we consider operators $T$ and $S$ which satisfy $\omega(T + \lambda S) = \omega(T)+\omega(S)$ for some complex unit $\lambda$. We show that $T \parallel_{\omega} S$ if and only if there exists a sequence of unit vectors $\{x_n\}$ in $\mathscr{H}$ such that \begin{align*} \lim_{n\rightarrow\infty} \big|\langle Tx_n, x_n\rangle\langle Sx_n, x_n\rangle\big| = \omega(T)\omega(S). \end{align*} We then apply it to give some applications.