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 $(\mathcal{H},⟨\cdot ,\cdot ⟩)$ 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 $\mathcal{H}$ such that \begin{align*} \lim_{n\rightarrow\infty} \big|\langle Tx_n, x_n\rangle\langle Sx_n, x_n\rangle\big| = ω(T)ω(S). \end{align*} We then apply it to give some applications.