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 $T\perp^{\varepsilon}_ω S$ if $$ω^2(T+λS)\geq ω^2(T)-2\varepsilon ω(T) ω(λS)\,\,\, \text{for all }λ\in\mathbb{C}.$$ We show that $T\perp^{\varepsilon}_ω S$ if and only if $\displaystyle\inf_{θ\in [0, 2π)} D^θ_ω(T, S) \geq -\varepsilon ω(T) ω(S)$ in which $D^θ_ω(T, S)=\displaystyle\lim_{r\to 0^+} \frac{ω^2(T+re^{iθ} S)-ω^2(T)}{2r}$; and this occurs if and only if for every $θ\in[0,2π)$, there exists a sequence $\{x_n^θ\}$ of unit vectors in $\mathscr{H}$ such that $$\displaystyle\lim_{n\to \infty} |\langle Tx^θ_n, x^θ_n\rangle|=ω(T),\,\, \text{and}\,\, \displaystyle\lim_{n\to \infty} {\rm Re}\{e^{-iθ} \langle Tx^θ_n, x^θ_n\rangle\bar{\langle Sx^θ_n, x^θ_n\rangle}\}\geq -\varepsilon ω(T) ω(S),$$ where $ω(T)$ is the numerical radius of $T$.