论文标题
伪$ s $ - Quaternionic Hilbert Space的特殊运营商的Spectra
Pseudo $S$-spectra of special operators in quaternionic Hilbert spaces
论文作者
论文摘要
对于一个有界的Quaternionic操作员$ t $在正确的Quaternionic Hilbert Space $ \ Mathcal {h} $和$ \ varepsilon> 0 $，$ t $的pseudo $ s $ s prectrum of $ t $定义为\ begin \ begin \ begin {align*} a align*}λ_ {λ_{λ_{\ varepsiLON} = \ bigCup \ left \ {q \ in \ mathbb {h} \setMinusσ_s（t）：\; \ |δ_{q}（t）^{ - 1} \ | \ geq \ frac {1} {\ varepsilon} \ right \}，\ end {align*}其中$ \ mathbb {h} $表示quaternions of quaternions ring of quaternions，$σ_s（t）$是$ s $ s $ s $ $ s $ $ t $ $ t $ and $ t $和$ t $和$ t $和$δ_Q（t） \ text {re}（q）t+| q |^2i $。这是从复杂的希尔伯特空间理论中对伪谱的自然概括。在本文中，我们研究了伪$ s $ spectrum的几个属性，并明确计算了一些特殊类别的运营商，例如上三角矩阵，自我伴随操作员，普通操作员，正常操作员和正交项目。 In particular, by an application of $S$-functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number $r$ if and only if for every $\varepsilon>0$ the pseudo $S$-spectrum of the operator is the circularization of a closed disc in the complex plane centered at $r$ with the radius $ \ sqrt {\ varepsilon} $。此外，我们为Quaternionic操作员提出了$ G_1 $的条件，并在此设置中证明了一些结果。
For a bounded quaternionic operator $T$ on a right quaternionic Hilbert space $\mathcal{H}$ and $\varepsilon >0$, the pseudo $S$-spectrum of $T$ is defined as \begin{align*} Λ_{\varepsilon}^{S}(T) := σ_S (T) \bigcup \left \{ q \in \mathbb{H}\setminus σ_S(T):\; \|Δ_{q}(T)^{-1}\| \geq \frac{1}{\varepsilon} \right\}, \end{align*} where $\mathbb{H}$ denotes the division ring of quaternions, $σ_S(T)$ is the $S$-spectrum of $T$ and $Δ_q(T)= T^2-2 \text{Re}(q)T+|q|^2I$. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate several properties of the pseudo $S$-spectrum and explicitly compute the pseudo $S$-spectra for some special classes of operators such as upper triangular matrices, self adjoint-operators, normal operators and orthogonal projections. In particular, by an application of $S$-functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number $r$ if and only if for every $\varepsilon>0$ the pseudo $S$-spectrum of the operator is the circularization of a closed disc in the complex plane centered at $r$ with the radius $\sqrt{\varepsilon}$. Further, we propose a $G_1$-condition for quaternionic operators and prove some results in this setting.
