Approximate numerical radius orthogonality

Abstract

We introduce the notion of approximate numerical radius (Birkhoff) orthogonality and investigate its significant properties. Let $T,S\in \mathbb{B}(\mathcal{H})$ and $\epsilon \in [0,1)$. We say that $T$ is approximate numerical radius orthogonal to $S$ and we write $T{\perp }_{\omega }^{\epsilon }S$ if $${\omega }^2(T+\lambda S)\ge {\omega }^2(T)-2\epsilon \omega (T)\omega (\lambda S)\text{for all }\lambda \in \mathbb{C}.$$ We show that $T{\perp }_{\omega }^{\epsilon }S$ if and only if $\underset{\theta \in [0,2\pi )}{\inf }{D}_{\omega }^{\theta }(T,S)\ge -\epsilon \omega (T)\omega (S)$ in which $D_{\omega }^{\theta }(T,S)=\underset{r\to 0^{+}}{\lim }\frac{{\omega }^2(T+r{e}^{i\theta }S)-{\omega }^2(T)}{2r}$; 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 $\mathcal{H}$ such that $$\underset{n\to \infty }{\lim }|⟨T{x}_n^{\theta },x_n^{\theta }⟩|=\omega (T),\text{and}\underset{n\to \infty }{\lim }\mathrm{Re}\{e^{-i\theta }⟨T{x}_n^{\theta },x_n^{\theta }⟩\overline{⟨S{x}_n^{\theta },x_n^{\theta }⟩}\}\ge -\epsilon \omega (T)\omega (S),$$ where $\omega (T)$ is the numerical radius of $T$.

14 pages