Abstract
In this chapter, we will present a proof of Loewner’s theorem due to Hansen–Pedersen that relies on the Krein–Milman theorem; we follow a variant of Hansen [133].
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References
D. H. Armitage, The Riesz-Herglotz representation for positive harmonic functions via Choquet’s theorem, Potential Theory—ICPT 94 (Kouty, 1994), pp. 229–232, de Gruyter, Berlin, 1996.
R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, 169, Springer-Verlag, New York, 1997.
L. Boutet de Monvel, unpublished note, 1989; included as an appendix to this book.
G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1955), 131–295.
G. Choquet, Lectures on Analysis, Volume I: Integration and Topological Vector Spaces; Volume II: Representation Theory; Volume III: Infinite Dimensional Measures and Problem Solutions, W.A. Benjamin, New York-Amsterdam, 1969.
F. Hansen, The fast track to Löwner’s theorem, Linear Algebra Appl. 438 (2013), 4557–4571.
F. Hansen and G. K. Pedersen, Jensen’s inequality for operators and Löwner’s theorem, Math. Ann. 258 (1982), 229–241.
F. Hiai and D. Petz, Introduction to Matrix Analysis and Applications, Springer, 2014.
F. Holland, The extreme points of a class of functions with positive real part, Math. Ann. 202 (1973), 85–87.
J. L. Kelley, Note on a theorem of Krein and Milman, J. Osaka Inst. Sci. Tech. Part I, 3 (1951), 1–2.
M. Krein and D. Milman, On extreme points of regularly convex sets, Studia Math. 9 (1940), 133–138.
J. Lindenstrauss, G. H. Olsen, and Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier Grenoble 28 (1978), 91–114.
R. R. Phelps, Integral representations for elements of convex sets, Studies in Functional Analysis, pp. 115–157, MAA Stud. Math., 21, Math. Assoc. America, Washington, DC, 1980.
E. T. Poulsen, A simplex with dense extreme boundary, Ann. Inst. Fourier Grenoble 11 (1961), 83–87.
B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. in Math. 137 (1998), 82–203.
B. Simon, Convexity: An Analytic Viewpoint, Cambridge Tracts in Mathematics, 187, Cambridge University Press, Cambridge, 2011.
B. Simon A Comprehensive Course in Analysis, Part 1: Real Analysis, American Mathematical Society, Providence, RI, 2015.
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Simon, B. (2019). The Krein–Milman Theorem and Hansen’s Variant of the Hansen–Pedersen Proof. In: Loewner's Theorem on Monotone Matrix Functions. Grundlehren der mathematischen Wissenschaften, vol 354. Springer, Cham. https://doi.org/10.1007/978-3-030-22422-6_28
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DOI: https://doi.org/10.1007/978-3-030-22422-6_28
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