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

2011/09/17 by Joseph A. Ball, Ball, Joseph A., Moisés Guerra Huamán +1
Mathematics · #46A55 #47A20 #47A48 #47A56 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:46A55 #msc:47A20 #msc:47A48 #msc:47A56

paper · pdf · doi:10.48550/arxiv.1109.3793

33 pages

arxiv created 2011/09/17 · arxiv updated 2011/09/20

Abstract

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.

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