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Journal of Operator Theory

Volume 70, Issue 2, Autumn 2013  pp. 531-571.

Convexity analysis and the matrix-valued Schur class over finitely connected planar domains

Authors J.A. Ball (1) and M.D. Guerra Huaman (2)
Author institution: (1) Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, U.S.A.
(2) Cal. Jose A. Quinones 135, Urb. Lucyana, Dist. Carabayllo Lima, Lima 6, Peru

Summary:  We identify the set of extreme points and apply Choquet theory to a normalized matrix-measure ball subject to finitely many linear side constraints. As an application we obtain integral representation formulas for the Herglotz class of matrix-valued functions on a finitely-connected planar domain and associated continuous Agler decompositions for the matrix-valued Schur class over the domain. The results give some additional insight into the negative answer to the spectral set problem over such domains recently obtained by Agler--Harland--Raphael and Dritschel--McCullough.

Keywords:  Choquet theory, positive operator measures, Schur class, finitely connected planar domain, $C^{*}$-convex combination, interior point of the $C^{*}$-convex hull

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