Numerical radius parallelism of Hilbert space operators

TL;DR

This paper introduces a new concept of parallelism for Hilbert space operators based on numerical radius, characterizes it via sequences of unit vectors, and explores its applications.

Contribution

It defines a novel numerical radius-based parallelism for operators and provides a characterization theorem with applications.

Findings

Operator parallelism characterized by sequences of unit vectors.

Equivalent condition involving numerical radius and inner products.

Applications demonstrating the usefulness of the new parallelism concept.

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.