Maps on matrices that preserve the spectrum

Let T be a continuous map of the space of complex n×n matrices into itself satisfying T(0)=0 such that the spectrum of T(x)-T(y) is always a subset of the spectrum of x-y. There exists then an invertible n×n matrix u such that either T(a)=uau^-1 for all a or T(a)=ua^tu^-1 for all a. We arrive at the same conclusion by supposing that the spectrum of x-y is always a subset of the spectrum of T(x)-Tt(y), without the continuity assumption on T.