Abstract. Let $\mn $ denote the algebra of all $n\times n$ complex matrices, and fix a nonzero vector $x_0$ in $\C ^n$. For any matrix $T\in \mn $, let $\sigma _T(x_0)$ denote the local spectrum of $T$ at $x_0$. Given three scalars $\mu ,~\nu $ and $\xi $ simultaneously nonzero, we study maps $\varphi $ on $\mn $ satisfying $ \sigma _{\mu STS + \nu T S +\xi ST}(x_0)= \sigma _{\mu \varphi (S)\varphi (T)\varphi (S) + \nu \varphi (T)\varphi (S)+\xi \varphi (S)\varphi (T)}(x_0) $ for all $S,~T\in \mn $. Our main result extends and unifies the main results of several papers on maps on $\mn $ preserving the local spectrum of different products.

DOI: 10.14232/actasm-017-590-0

AMS Subject Classification (1991): 47B49; 47A10, 47A11

Keyword(s): nonlinear preservers, local spectrum, SVEP, quadratic product, Jordan product, matrices

Received December 30, 2017 and in final form March 8, 2018. (Registered under 90/2017.)