Convexity and star-shapedness of matricial range

Let A=(A1,…,Am) be an m-tuple of bounded linear operators acting on a Hilbert space H. Their joint (p,q)-matricial range Λp,q(A) is the collection of (B1,…,Bm)∈Mqm, where Ip⊗Bj is a compression of Aj 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 Λp,q(A) is star-shaped if the dimension of H is sufficiently large. If dim⁡H is infinite, we extend the definition of Λp,q(A) to Λ∞,q(A) consisting of (B1,…,Bm)∈Mqm such that I∞⊗Bj is a compression of Aj on a closed subspace of H, and consider the joint essential (p,q)-matricial rangeΛp,qess(A)=⋂{cl(Λp,q(A1+F1,…,Am+Fm)):F1,…,Fm are compact operators}. Both sets are shown to be convex, and the latter one is always non-empty and compact.