Joint matricial range and joint congruence matricial range of operators

Let 𝐀= (A_1, … , A_m) , where A_1, … , A_m are n× n real matrices. The real joint ( p , q )-matricial range of 𝐀 , ^ℝ_p,q(𝐀) , is the set of m -tuple of q× q real matrices (B_1, … , B_m) such that (X^*A_1X, … , X^*A_mX) = (I_p⊗ B_1, … , I_p ⊗ B_m) for some real n × pq matrix X satisfying X^*X = I_pq . It is shown that if n is sufficiently large, then the set ^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.