Let \({\mathscr {L}}({\mathscr {H}})\) be the algebra of all bounded linear operators on a complex Hilbert space \({\mathscr {H}}\) with \(\dim {\mathscr {H}}\ge 3\), and let \(\mathscr {A} \) and \(\mathscr {B}\) be two subsets of \({\mathscr {L}}({\mathscr {H}})\) containing all operators of rank at most one. For \(\varepsilon \in (0,1)\) the \(\varepsilon \)-condition spectrum of any \(A\in {\mathscr {L}}({\mathscr {H}})\) is defined by$$\begin{aligned} \sigma _{\epsilon }(A) := \sigma (A)\cup \left\{ \lambda \in \mathbb {C}\setminus \sigma (A):~\Vert (\lambda I -A)^{-1}\Vert \Vert \lambda I -A\Vert \ge \frac{1}{\varepsilon }\right\} , \end{aligned}$$where \(\sigma (A)\) is the spectrum of A. The \(\varepsilon \)-condition spectral radius of A is given by$$\begin{aligned} r_\varepsilon (A):=\sup \left\{ |z| : z\in \sigma _\varepsilon (A) \right\} . \end{aligned}$$We compute the \(\varepsilon \)-condition spectrum of any operator of rank at most one, and give an explicit formula for its \(\varepsilon \)-condition spectral radius. It is then illustrated that the results can be applied to characterize surjective mappings \(\phi :\mathscr {A} \longrightarrow \mathscr {B}\) satisfying$$\begin{aligned} \delta (\phi (A)^*\phi (B)) = \delta (A^*B) \quad \text{ for } \text{ all } A,B\in \mathscr {A} \end{aligned}$$where \(\delta \) stands for \(\sigma _\varepsilon (\cdot )\) or \(r_\varepsilon (\cdot ).\)

中文翻译：

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一阶算子的条件谱和算子的偏积条件谱的保存者

令\（{\ mathscr {L}}（{\ mathscr {H}}）\）是带有\（\ dim的复数希尔伯特空间\（{\ mathscr {H}} \\）上所有有界线性算子的代数{\ mathscr {H}} \ ge 3 \），然后让\（\ mathscr {A} \）和\（\ mathscr {B} \）是\（{\ mathscr {L}}（{\ mathscr {H}}）\）包含所有排名最高的运算符。对于\（\ varepsilon \ in（0,1）\），任何\（A \ in {\ mathscr {L}}（{\ mathscr {H}}）\）的\（\ varepsilon \）条件频谱为被定义为$$ \ begin {aligned} \ sigma _ {\ epsilon}（A）：= \ sigma（A）\ cup \ left \ {\ lambda \ in \ mathbb {C} \ setminus \ sigma（A）：〜\ Vert （\ lambda I -A）^ {-1} \ Vert \ Vert \ lambda I -A \ Vert \ ge \ frac {1} {\ varepsilon} \ right \}，\ end {aligned} $$其中\（\ sigma（A）\）是A的频谱。的\（\ varepsilon \） -condition的谱半径阿由下式给出$$ \ {开始对准} R_ \ varepsilon（A）：= \ SUP \左\ {| Z | ：z \ in \ sigma _ \ varepsilon（A）\ right \}。\ end {aligned} $$我们计算任何排名最高的算子的\（\ varepsilon \）-条件谱，并为其\（\ varepsilon \）给出一个明确的公式条件光谱半径。然后说明可以将结果应用于表征满足$$ \ begin {aligned} \ delta（\ phi（A）^的射影映射\（\ phi：\ mathscr {A} \ longrightarrow \ mathscr {B} \）的特征* \ phi（B））= \ delta（A ^ * B）\ quad \ text {for} \ text {all} A，B \ in \ mathscr {A} \ end {aligned} $$其中\（\ delta \）代表\（\ sigma _ \ varepsilon（\ cdot）\）或\（r_ \ varepsilon（\ cdot）。\）