The joint essential numerical range of operators: convexity and related results

Let W(A) and W(e)(A) be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A(1),.., A(m)) acting on an infinite-dimensional Hilbert space. It is shown that W(e)(A) is always convex and admits many equivalent formulations. In particular, for any fixed i is an element of {1,..., m}, W(e) (A) can be obtained as the intersection of all sets of the form cl(W(A(1),..., A(i+1), A(i) + F, A(i+1),..., A(m))), where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in W(e)(A) as star centers. Although cl(W(A)) is usually not convex, an analog of the separation theorem is obtained, namely, for any element d is not an element of cl(W(A)), there is a linear functional f such that f (d) \u3e sup{f (a) : a is an element of cl(W((A) over tilde))}, where (A) over tilde is obtained from A by perturbing one of the components A(i) by a finite rank self-adjoint operator. Other results on W(A) and W(e)(A) extending those on a single operator axe obtained