Approximate numerical radius orthogonality

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)$\mathrm{.}Wesaythat$T$ is approximate numerical radius orthogonal to $S$ and we write $T{\perp }_{\omega }^{\epsilon }S$ if $${\omega }^2(T+\lambda S)\ge {\omega }^2(T)-2\epsilon \omega (T)\omega (\lambda S)\text{for all }\lambda \in \mathbb{C}\mathrm{.}$$ We show that $T{\perp }_{\omega }^{\epsilon }S$ if and only if $\displaystyle\inf_{\theta\in [0, 2\pi)} D^{\theta}{\omega}(T, S) \geq -\varepsilon \omega(T) \omega(S)$inwhich$D^{\theta}{\omega}(T, S)=\displaystyle\lim_{r\to 0^+} \frac{\omega^2(T+re^{i\theta} S)-\omega^2(T)}{2r}$;andthisoccursifandonlyifforevery$\theta\in[0,2\pi)$,thereexistsasequence${x_n^{\theta}}$ of unit vectors in $\mathcal{H}$ such that $${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\mathrm{\mid }⟨T{x}_n^{\theta },x_n^{\theta }⟩\mathrm{\mid }=\omega (T),\text{and}{\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\mathrm{R}\mathrm{e}\{e^{-i\theta }⟨T{x}_n^{\theta },x_n^{\theta }⟩\overline{⟨S{x}_n^{\theta },x_n^{\theta }⟩}\}\ge -\epsilon \omega (T)\omega (S),}}$$ where $\omega (T)$ is the numerical radius of $T$.