Joint matricial range and joint congruence matricial range of operators

Let A = (A(1), ..., A(m)), where A(1), ..., A(m) are n x n real matrices. The real joint (p, q)-matricial range of A, Lambda(R)(p,q) (A), is the set of m-tuple of q x q real matrices (B-1, ..., B-m) such that (X* A(1)X, ..., X* A(m) X) = (I-p circle times B-1, ..., I-p circle times B-m) for some real n x pq matrix X satisfying X* X = I-pq. It is shown that if n is sufficiently large, then the set Lambda(R)(p,q) (A) is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space H, and used to show that the joint essential (p, q)-matricial range of A is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.