The Joint Essential Numerical Range of operators: Convexity and Related Results
نویسندگان
چکیده
LetW (A) andWe(A) be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A1, . . . , Am) acting on an infinite dimensional Hilbert space, respectively. In this paper, it is shown that We(A) is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ {1, . . . ,m}, We(A) can be obtained as the intersection of all sets of the form cl(W (A1, . . . , Ai+1, Ai + F,Ai+1, . . . , Am)), where F = F ∗ has finite rank Moreover, it is shown that the closure cl(W (A)) of W (A) is always star-shaped with the elements in We(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 / ∈ cl(W (A)), there is a linear functional f such that f(d) > sup{f(a) : a ∈ cl(W (Ã))}, where à is obtained from A by perturbing one of the components Ai by a finite rank self-adjoint operator. Other results on W (A) and We(A) extending those on a single operator are obtained. AMS Subject Classification 47A12, 47A13, 47A55.
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