# Positive Definite Functions on Infinite-Dimensional Convex Cones

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*Helge Glöckner*

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.

#### Table of Contents

# Table of Contents

## Positive Definite Functions on Infinite-Dimensional Convex Cones

- Contents v6 free
- Introduction ix10 free
- Part I. Preliminaries and Preparatory Results 116 free
- 1 Bounded and unbounded operators 116
- 2 Cone-valued measures 621
- 3 Measures on topological spaces 1631
- 4 Projective limits of cone-valued measures 1934
- 5 Holomorphic functions 2540
- 6 Involutive semigroups and their representations 2742
- 7 Positive definite kernels and functions 2944
- 8 C*-algebras associated with involutive semigroups 3954
- 9 Integral representations of positive definite functions 4156
- 10 Convex cones and their faces 4560
- 11 Examples of convex cones 5166
- 12 Conelike semigroups: definition and examples 5368
- 13 Representations of conelike semigroups I 5570
- 14 Fourier and Laplace transforms 5873
- 15 Generalized Bochner and Stone Theorems 6378

- Part II. Main Results 6782
- 16 Nussbaum Theorem for open convex cones 6782
- 17 Positive definite functions on convex cones with non-empty interior 7489
- 18 Positive definite functions on convex sets 8095
- 19 Associated Hilbert spaces and representations 88103
- 20 Nussbaum Theorem for generating convex cones 98113
- 21 Representations of conelike semigroups II 105120
- 22 Associated unitary representations 108123
- 23 Holomorphic extension of unitary representations 110125
- 24 Holomorphic extension of representations of nuclear groups 113128

- References 115130
- Index 123138 free
- List of Symbols 127142