Approximate numerical radius orthogonality

TL;DR

This paper introduces and studies the concept of approximate numerical radius orthogonality for operators, providing characterizations and conditions involving sequences of vectors and derivatives of the numerical radius.

Contribution

It defines approximate numerical radius orthogonality and characterizes it through derivative limits and vector sequences, extending the understanding of operator orthogonality.

Findings

Characterization of approximate numerical radius orthogonality via derivatives.

Equivalence with existence of specific vector sequences.

Provides conditions involving the numerical radius and operator sequences.

Abstract

We introduce the notion of approximate numerical radius (Birkhoff) orthogonality and investigate its significant properties. Let $T, S\in \mathbb{B}(\mathscr{H})$ and $\varepsilon \in [0, 1)$. We say that $T$ is approximate numerical radius orthogonal to $S$ and we write $T\perp^{\varepsilon}_{\omega} S$ if $$\omega^2(T+\lambda S)\geq \omega^2(T)-2\varepsilon \omega(T) \omega(\lambda S)\,\,\, \text{for all }\lambda\in\mathbb{C}.$$ We show that $T\perp^{\varepsilon}_{\omega} S$ if and only if $\displaystyle\inf_{\theta\in [0, 2\pi)} D^{\theta}_{\omega}(T, S) \geq -\varepsilon \omega(T) \omega(S)$ in which $D^{\theta}_{\omega}(T, S)=\displaystyle\lim_{r\to 0^+} \frac{\omega^2(T+re^{i\theta} S)-\omega^2(T)}{2r}$; and this occurs if and only if for every $\theta\in[0,2\pi)$, there exists a sequence $\{x_n^{\theta}\}$ of unit vectors in $\mathscr{H}$ such that $$\displaystyle\lim_{n\to \infty} |\langle Tx^{\theta}_n, x^{\theta}_n\rangle|=\omega(T),\,\, \text{and}\,\, \displaystyle\lim_{n\to \infty} {\rm Re}\{e^{-i\theta} \langle Tx^{\theta}_n, x^{\theta}_n\rangle\bar{\langle Sx^{\theta}_n, x^{\theta}_n\rangle}\}\geq -\varepsilon \omega(T) \omega(S),$$ where $\omega(T)$ is the numerical radius of $T$.