Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on $\mathcal{H}$. For $A\in \mathcal{B}(\mathcal{H})$, let ${\sigma }_{\ast }(A)$ denote the semi-Fredholm domain, the Fredholm domain or the Weyl domain of A in the spectrum, σ(A). We characterize the form of a map ϕ from $\mathcal{B}(\mathcal{H})$ into itself with range contains the operators of rank at most two and satisfies ${\sigma }_{\ast }(\varphi (A)\varphi (B{)}^{\ast })={\sigma }_{\ast }(A{B}^{\ast })$ for all $A,B\in \mathcal{B}(\mathcal{H})$. We also describe those maps on $\mathcal{B}(\mathcal{H})$ preserving the same spectral set ${\sigma }_{\ast }(.)$ ‘but of Jordan skew-triple product’ $A{B}^{\ast }A$ ‘of operators’.

中文翻译：

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保留算子偏斜积的一些谱域的地图

让$\mathcal{H}$是一个无限维的希尔伯特空间，并且$\mathcal{乙}(\mathcal{H})$是所有有界线性算子的代数$\mathcal{H}$. 为了$A\in \mathcal{乙}(\mathcal{H})$， 让${\sigma }_{＊}(A)$表示谱中A的半 Fredholm 域、Fredholm 域或 Weyl 域， σ ( A )。我们从中表征映射ϕ的形式$\mathcal{乙}(\mathcal{H})$进入其自身，其范围包含最多两个等级的运算符并满足${\sigma }_{＊}(\phi (A)\phi (乙{)}^{＊})={\sigma }_{＊}(A{乙}^{＊})$对全部$A,乙\in \mathcal{乙}(\mathcal{H})$. 我们还描述了这些地图$\mathcal{乙}(\mathcal{H})$保留相同的光谱集${\sigma }_{＊}(.)$'但乔丹偏斜三重产品'$A{乙}^{＊}A$'运营商'。