Maps preserving the local spectrum of quadratic products of matrices

Let Mn(C) denote the algebra of all n × n complex matrices, and fix a nonzero vector x0 in C n . For any matrix T ∈ Mn(C), let σT (x0) denote the local spectrum of T at x0. Given three scalars µ, ν and ξ simultaneously nonzero, we study maps ϕ on Mn(C) satisfying σµST S+νT S+ξST (x0) = σµϕ(S)ϕ(T )ϕ(S)+νϕ(T )ϕ(S)+ξϕ(S)ϕ(T )(x0) for all S, T ∈ Mn(C). Our main result extends and unifies the main results of several papers on maps on Mn(C) preserving the local spectrum of different products