Generalized matricial ranges and positive definiteness

Numerical range is the subject of study for over a century. Its extension to matricial range is defined in terms of completely positive maps. A linear map ϕ between matrix algebras is completely positive if and only if its Choi matrix C(ϕ) is positive semidefinite. In this paper, we extend the study to some joint matricial ranges defined by those ϕ's where C(ϕ) is Hermitian with specified spectrum. We prove some convexity theorems which extend previous results on 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.