Convexity and star-shapedness of matricial range

Let A=(A ₁,…,A _m) be an m-tuple of bounded linear operators acting on a Hilbert space H. Their joint (p,q)-matricial range Λ _p,q(A) is the collection of (B ₁,…,B _m)∈M _q ^m, where I _p⊗B _j is a compression of A _j on a pq-dimensional subspace. This definition covers various kinds of generalized numerical ranges for different values of p,q,m. In this paper, it is shown that Λ _p,q(A) is star-shaped if the dimension of H is sufficiently large. If dim⁡H is infinite, we extend the definition of Λ _p,q(A) to Λ _∞,q(A) consisting of (B ₁,…,B _m)∈M _q ^m such that I _∞⊗B _j is a compression of A _j on a closed subspace of H, and consider the joint essential (p,q)-matricial range Λ _p,q ^ess(A)=⋂{cl(Λ _p,q(A ₁+F ₁,…,A _m+F _m)):F ₁,…,F _m are compact operators}. Both sets are shown to be convex, and the latter one is always non-empty and compact.