Joint matricial range and joint congruence matricial range of operators

Abstract

Let A= (A1, … , Am) , where A1, … , Am are n× n real matrices. The real joint (p, q)-matricial range of A, Λp,qR(A), is the set of m-tuple of q× q real matrices (B1, … , Bm) such that (X∗A1X, … , X∗AmX) = (Ip⊗ B1, … , Ip⊗ Bm) for some real n× pq matrix X satisfying X∗X= Ipq. It is shown that if n is sufficiently large, then the set Λp,qR(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.