Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-10-27 , DOI: 10.1007/s00009-020-01644-x Constantin Costara
Let X be a complex Banach space and let \(x_{0}\in X\) be a fixed nonzero vector. Denote by \(\mathcal {L}\left( X\right) \) the algebra of all linear and bounded operators on X, and for \(T \in \mathcal {L}\left( X\right) \) denote by \(\sigma _{T}\left( x_{0}\right) \) the local spectrum of T at \(x_{0}\). We characterize linear and surjective maps \(\varphi :\mathcal {L} \left( X\right) \rightarrow \mathcal {L}\left( X\right) \) such that \(\varphi \left( I\right) \in \mathcal {L}\left( X\right) \) is invertible and
$$\begin{aligned} 0\in \sigma _{T}\left( x_{0}\right) \Longleftrightarrow 0\in \sigma _{\varphi \left( T\right) }\left( x_{0}\right) \qquad \left( T\in \mathcal {L}\left( X\right) \right) . \end{aligned}$$.
中文翻译:
在某些固定矢量处保持局部谱半径为零的线性映射算子
令X为复Banach空间,令\(x_ {0} \ in X \)为固定的非零向量。表示由\(\ mathcal {L} \左(X \右)\)的所有线性的代数和上界运营商X,以及用于\(T \在\ mathcal {L} \左(X \右)\)用\(\ sigma _ {T} \ left(x_ {0} \ right)\)表示T在\(x_ {0} \)的局部频谱。我们刻画线性映射和射影映射\(\ varphi:\ mathcal {L} \ left(X \ right)\ rightarrow \ mathcal {L} \ left(X \ right)\)使得\(\ varphi \ left(I \ right)\ in \ mathcal {L} \ left(X \ right)\)是可逆的,
$$ \ begin {aligned} 0 \ in \ sigma _ {T} \ left(x_ {0} \ right)\ Longleftrightarrowarrow 0 \ in \ sigma _ {\ varphi \ left(T \ right)} \ left(x_ { 0} \ right)\ qquad \ left(T \ in \ mathcal {L} \ left(X \ right)\ right)。\ end {aligned} $$。