Additive Maps Preserving Local Spectrum

Abstract.

Let X be a complex Banach space, and let $\mathcal{L}(X)$ be the space of bounded operators on X. Given $T \in \mathcal{L}(X)$ and x ∈ X, denote by σ _T (x) the local spectrum of T at x.

We prove that if $\Phi :\mathcal{L}(X) \to \mathcal{L}(X)$ is an additive map such that

$$ \sigma _{{\Phi (T)}} (x) = \sigma _{{T(x)}} \quad (T \in \mathcal{L}(x),\;x \in X), $$

then Φ (T)  =  T for all $T \in \mathcal{L}(X).$ We also investigate several extensions of this result to the case of $\Phi :\mathcal{L}(X) \to \mathcal{L}(Y),$ where $X \ne Y.$

The proof is based on elementary considerations in local spectral theory, together with the following local identity principle: given $S,T \in \mathcal{L}(X)$ and x ∈X, if σ_S+R (x)  =  σ_T+R (x) for all rank one operators $R \in \mathcal{L}(X),$ then S _x   =  T _x .