论文标题
近似数值半径正交性
Approximate numerical radius orthogonality
论文作者
论文摘要
我们介绍了近似数值半径（Birkhoff）正交性的概念，并研究了其重要特性。令$ t，s \ in \ mathbb {b}（\ mathscr {h}）$和$ \ varepsilon \ in [0，1）$。我们说$ t $是近似的数值半径正交到$ s $，并且我们写$ t \ perp^{\ varepsilon}_ΩS$如果$$ω^2（t+λss）\ geq ocqω^2（t）-2 \ varepsilonω }λ\ in \ Mathbb {C}。$$我们表明$ t \ perp^{\ varepsilon}_Ωs $ if and仅当$ \ displayStyle \ inplayStyle \ inf_ {θ\ in [0，2π）} d^θ_Ω $ d^θ_为（t，s）= \ displaystyle \ lim_ {r \ to 0^+} \ frac {ω^2（t+re^{iθ} s）-Ω^2（t）}} {2r} $;并且仅在[0,2π）$中的每一个$θ\时就会发生这种情况。 x^θ_n\ rangle | =ω（t），\，\，\ text {and} \，\，\，\ displayStyle \ lim_ {n \ to \ infty} {\ rm re} {\ rm re} sx^θ_n，x^θ_n\ rangle} \ geq- \ geq - \varepsilonΩ（t）ω（s），$$，其中$ω（t）$是$ t $的数值半径。
We introduce the notion of approximate numerical radius (Birkhoff) orthogonality and investigate its significant properties. Let $T, S\in \mathbb{B}(\mathscr{H})$ and $\varepsilon \in [0, 1)$. We say that $T$ is approximate numerical radius orthogonal to $S$ and we write $T\perp^{\varepsilon}_ω S$ if $$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 $T$.
