We describe linear maps from a C * -algebra onto another one preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the reduced minimum modulus.

Abstract. Let x0 be a nonzero vector in C n . We show that a linear map Φ : Mn(C) → Mn(C) preserves the local spectral radius at x0 if and only if there is α ∈ C of modulus one and an invertible matrix A ∈ Mn(C) such that Ax0 = x0 and Φ(T ) = αAT A −1 for all T ∈ Mn(C).

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Let X be a complex Banach space, and let L(X) be the space of bounded operators on X. Given T ∈ L(X) and x ∈ X, denote by σT (x) the local spectrum of T at x.We prove that if Φ : L(X) → L(X) is an additive map such thatthen Φ(T ) = T for all T ∈ L(X). We also investigate several extensions of this result to the case of Φ :The proof is based on elementary considerations in local spectral theory, together with the following local identity principle: given S, T ∈ L(X) andx ∈ X, if σS+R(x) = σT +R (x) for all rank one operators R ∈ L(X), then Sx = T x. Mathematics Subject Classification (2000). Primary 47A11; Secondary 47A10, 47B48.

Abstract. We study the local spectral properties of both unilateral and bilateral weighted shift operators.