ABSTRACT

Let $\mathcal{L}\left(X\right)$ be the Banach algebra of all bounded linear operators on a complex Banach space X. Let $X_T\left(\right\{\lambda \left\}\right)$ denote the local spectral subspace of an operator $T\in \mathcal{L}\left(X\right)$ associated with a singleton $\left\{\lambda \right\}\subset \mathbb{C}$, and fix a scalar ${\lambda }_0\in \mathbb{C}$. We characterize maps ϕ on $\mathcal{L}\left(X\right)$ which satisfy $X_{\varphi \left(T\right)-\varphi \left(S\right)}\left(\right\{{\lambda }_0\left\}\right)=X_{T-S}\left(\right\{{\lambda }_0\left\}\right)\text{for all }T,S\in \mathcal{L}\left(X\right).$Furthermore, we give a characterization of maps ϕ on $\mathcal{L}\left(X\right)$ for which $\varphi \left(0\right)=0$ and $X_{\varphi \left(T\right)+\varphi \left(S\right)}\left(\right\{{\lambda }_0\left\}\right)=X_{T+S}\left(\right\{{\lambda }_0\left\}\right)\text{for all }T,S\in \mathcal{L}\left(X\right).$

摘要

让 $\mathcal{大号}（X）$是复Banach空间X上所有有界线性算子的Banach代数。让$X_{Ť}（\left\{\lambda \right\}）$ 表示算子的局部频谱子空间 $Ť\in \mathcal{大号}（X）$ 与单身人士相关 $\left\{\lambda \right\}\subset \mathbb{C}$，并修复标量 ${\lambda }_0\in \mathbb{C}$。我们在上表征地图ϕ$\mathcal{大号}（X）$ 满足 $X_{\varphi （Ť）-\varphi （\mathrm{小号}）}（\left\{{\lambda }_0\right\}）=X_{Ť-\mathrm{小号}}（\left\{{\lambda }_0\right\}）\text{对全部 }Ť，\mathrm{小号}\in \mathcal{大号}（X）。$此外，我们给出一个定性的地图φ上$\mathcal{大号}（X）$ 为此 $\varphi （0）=0$ 和 $X_{\varphi （Ť）+\varphi （\mathrm{小号}）}（\left\{{\lambda }_0\right\}）=X_{Ť+\mathrm{小号}}（\left\{{\lambda }_0\right\}）\text{对全部 }Ť，\mathrm{小号}\in \mathcal{大号}（X）。$