Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector

In this paper, we characterize linear maps on matrix spaces that preserve matrices of local spectral radius zero at some fixed nonzero vector.

The local spectral radius of a nonnegative orbit of compact linear operators

We consider orbits of compact linear operators in a real Banach space which are nonnegative with respect to the partial ordering induced by a given cone. The main result shows that under a mild additional assumption the local spectral radius of a nonnegative orbit is an eigenvalue of the operator with a positive eigenvector.