Spectral properties of hermitian-preserving maps, generalized matricial ranges, and positive definiteness

Abstract

The numerical range has been a subject of study for over a century. Over the years, this object has seen several generalizations including a matricial extension defined in terms of completely positive maps. A linear map $\phi$ between matrix algebras is completely positive if and only if its Choi matrix $C(\phi)$ is positive semidefinite. In this dissertation, we give a unified treatment of the numerical and matricial ranges by extending to some joint matricial ranges defined by those maps $\phi$ for which $C(\phi)$ is a Hermitian matrix with a specific spectrum. We then prove some convexity theorems which extend previous results on the generalized joint numerical ranges and matricial ranges. We also extend Bohnenblust's result on joint positive definiteness of Hermitian matrices and Friedland and Loewy's result on the existence of a nonzero matrix with multiple first eigenvalue in subspaces of Hermitian matrices.