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Positive Definite Functions on Infinite-Dimensional Convex Cones
Helge Glöckner, Technische Universität Darmstadt, Germany
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Memoirs of the American Mathematical Society
2003; 128 pp; softcover
Volume: 166
ISBN-10: 0-8218-3256-5
ISBN-13: 978-0-8218-3256-1
List Price: US$62 Individual Members: US$37.20
Institutional Members: US\$49.60
Order Code: MEMO/166/789

This memoir is devoted to the study of positive definite functions on convex subsets of finite- or infinite-dimensional vector spaces, and to the study of representations of convex cones by positive operators on Hilbert spaces. Given a convex subset $$\Omega\subseteq V$$ of a real vector space $$V$$, we show that a function $$\phi\!:\Omega\to\mathbb{R}$$ is the Laplace transform of a positive measure $$\mu$$ on the algebraic dual space $$V^*$$ if and only if $$\phi$$ is continuous along line segments and positive definite. If $$V$$ is a topological vector space and $$\Omega\subseteq V$$ an open convex cone, or a convex cone with non-empty interior, we describe sufficient conditions for the existence of a representing measure $$\mu$$ for $$\phi$$ on the topological dual space$$V'$$. The results are used to explore continuity properties of positive definite functions on convex cones, and their holomorphic extendibility to positive definite functions on the associated tubes $$\Omega+iV\subseteq V_{\mathbb{C}}$$. We also study the interplay between positive definite functions and representations of convex cones, and derive various characterizations of those representations of convex cones on Hilbert spaces which are Laplace transforms of spectral measures. Furthermore, for scalar- or operator-valued positive definite functions which are Laplace transforms, we realize the associated reproducing kernel Hilbert space as an $$L^2$$-space $$L^2(V^*,\mu)$$ of vector-valued functions and link the natural translation operators on the reproducing kernel space to multiplication operators on $$L^2(V^*,\mu)$$, which gives us refined information concerning the norms of these operators.This memoir is devoted to the study of positive definite functions on convex subsets of finite- or infinite-dimensional vector spaces, and to the study of representations of convex cones by positive operators on Hilbert spaces. Given a convex subset $$\Omega\subseteq V$$ of a real vector space $$V$$, we show that a function $$\phi\!:\Omega\to\mathbb{R}$$ is the Laplace transform of a positive measure $$\mu$$ on the algebraic dual space $$V^*$$ if and only if $$\phi$$ is continuous along line segments and positive definite. If $$V$$ is a topological vector space and $$\Omega\subseteq V$$ an open convex cone, or a convex cone with non-empty interior, we describe sufficient conditions for the existence of a representing measure $$\mu$$ for $$\phi$$ on the topological dual space $$V'$$. The results are used to explore continuity properties of positive definite functions on convex cones, and their holomorphic extendibility to positive definite functions on the associated tubes $$\Omega+iV\subseteq V_\mathbb C$$. We also study the interplay between positive definite functions and representations of convex cones, and derive various characterizations of those representations of convex cones on Hilbert spaces which are Laplace transforms of spectral measures. Furthermore, for scalar- or operator-valued positive definite functions which are Laplace transforms, we realize the associated reproducing kernel Hilbert space as an $$L^2$$-space $$L^2(V^*,\mu)$$ of vector-valued functions and link the natural translation operators on the reproducing kernel space to multiplication operators on $$L^2(V^*,\mu)$$, which gives us refined information concerning the norms of these operators.

Graduate students and research mathematicians interested in analysis.

Part I. Preliminaries and Preparatory Results
• Bounded and unbounded operators
• Cone-valued measures
• Measures on topological spaces
• Projective limits of cone-valued measures
• Holomorphic functions
• Involutive semigroups and their representations
• Positive definite kernels and functions
• $$C^*$$-algebras associated with involutive semigroups
• Integral representations of positive definite functions
• Convex cones and their faces
• Examples of convex cones
• Conelike semigroups: definition and examples
• Representations of conelike semigroups I
• Fourier and Laplace transforms
• Generalized Bochner and Stone Theorems
Part II. Main Results
• Nussbaum Theorem for open convex cones
• Positive definite functions on convex cones with non-empty interior
• Positive definite functions on convex sets
• Associated Hilbert spaces and representations
• Nussbaum Theorem for generating convex cones
• Representations of conelike semigroups II
• Associated unitary representations
• Holomorphic extension of unitary representations
• Holomorphic extension of representations of nuclear groups
• References
• Index
• List of symbols