On spectrally bounded linear maps on [formula omitted

Abstract

Let X be a complex Banach space and denote by B(X) the algebra of all linear bounded operators on X. For T∈B(X), denote by r(T) its spectral radius. We prove that if φ:B(X)→B(X) is a linear and surjective map such thati)given any A,B∈B(X), there exists a positive real number MA,B such that r(φ(A)+λφ(B))≤MA,Br(A+λB) for all complex numbers λ satisfying |λ|≤1;ii)r(φ(T))=0 implies r(T)=0, then there exist a linear functional f:B(X)→C, a nonzero complex number c and either a bounded invertible linear map U:X→X such thatφ(T)=cUTU−1+f(T)I(T∈B(X)), or a bounded invertible linear map U:X⁎→X such thatφ(T)=cUT⁎U−1+f(T)I(T∈B(X)). Also, the linear functional f satisfies the property that given any A,B∈B(X) there exists NA,B≥0 such that |f(A)+λf(B)|≤NA,Br(A+λB) for all complex numbers λ satisfying |λ|≤1.