Approximate numerical radius orthogonality

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 $T$.