Joint matricial range and joint congruence matricial range of operators

Let $$\mathbf{A}= (A_1, \ldots , A_m)$$ , where $$A_1, \ldots , A_m$$ are $$n\times n$$ real matrices. The real joint (p, q)-matricial range of $$\mathbf{A}$$ , $${\varLambda }^{{\mathbb {R}}}_{p,q}(\mathbf{A})$$ , is the set of m-tuple of $$q\times q$$ real matrices $$(B_1, \ldots , B_m)$$ such that $$(X^*A_1X, \ldots , X^*A_mX) = (I_p\otimes B_1, \ldots , I_p \otimes B_m)$$ for some real $$n \times pq$$ matrix X satisfying $$X^*X = I_{pq}$$ . It is shown that if n is sufficiently large, then the set $${\varLambda }^{{\mathbb {R}}}_{p,q}(\mathbf{A})$$ is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space $${{{\mathcal {H}}}}$$ , and used to show that the joint essential (p, q)-matricial range of $$\mathbf{A}$$ is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.