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2009 | 194 | 1 | 91-104
Tytuł artykułu

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

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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 = (A₁, ..., Aₘ) 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 ∈ {1, ..., m}, $W_{e}(A)$ can be obtained as the intersection of all sets of the form
$cl(W(A₁, ..., A_{i+1}, A_{i} + F, A_{i+1}, ..., Aₘ))$,
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 ∉ 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 $A_{i}$ by a finite rank self-adjoint operator. Other results on W(A) and $W_{e}(A)$ extending those on a single operator are obtained.
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  • Department of Mathematics, The College of William and Mary, Williamsburg, VA 23185, U.S.A.
  • Department of Mathematics, Iowa State University, Ames, IA 50011, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm194-1-6
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