caa a22 4500 467931550 CHVBK 20180406152945.0 cr unu---uuuuu 170328e20060701xx s 000 0 eng 10.1007/s00020-005-1392-2 doi (NATIONALLICENCE)springer-10.1007/s00020-005-1392-2 Additive Maps Preserving Local Spectrum [Elektronische Daten] [Abdellatif Bourhim, Thomas Ransford] 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 σT (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 σS+R (x)=σT+R (x) for all rank one operators $$R \in \mathcal{L}(X),$$ then Sx=Tx . Birkhäuser Verlag, Basel, 2006 Local spectrum nationallicence single-valued extension property nationallicence Bourhim Abdellatif Département de mathématiques et de statistique, Université Laval, G1K 7P4, Québec (Québec), Canada aut Ransford Thomas Département de mathématiques et de statistique, Université Laval, G1K 7P4, Québec (Québec), Canada aut Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhauser.ch 55/3(2006-07-01), 377-385 0378-620X 55:3<377 2006 55 20 https://doi.org/10.1007/s00020-005-1392-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00020-005-1392-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bourhim Abdellatif Département de mathématiques et de statistique, Université Laval, G1K 7P4, Québec (Québec), Canada aut NATIONALLICENCE 700 1- Ransford Thomas Département de mathématiques et de statistique, Université Laval, G1K 7P4, Québec (Québec), Canada aut NATIONALLICENCE 773 0- Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhauser.ch 55/3(2006-07-01), 377-385 0378-620X 55:3<377 2006 55 20 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer