By Helge Glockner
This memoir is dedicated to the research of optimistic sure services on convex subsets of finite- or infinite-dimensional vector areas, and to the learn of representations of convex cones via confident operators on Hilbert areas. Given a convex subset $\Omega\sub V$ of a true vector area $V$, we exhibit functionality $\phi\!:\Omega\to\mathbb{R}$ is the Laplace rework of a good degree $\mu$ at the algebraic twin area $V^*$ if and provided that $\phi$ is continuing alongside line segments and confident yes. If $V$ is a topological vector area and $\Omega\sub V$ an open convex cone, or a convex cone with non-empty inside, we describe enough stipulations for the life of a representing degree $\mu$ for $\phi$ at the topological twin space$V$.The effects are used to discover continuity houses of confident yes capabilities on convex cones, and their holomorphic extendibility to confident sure features at the linked tubes $\Omega+iV\sub V_{\mathbb{C}}$. We additionally research the interaction among optimistic certain services and representations of convex cones, and derive a number of characterizations of these representations of convex cones on Hilbert areas that are Laplace transforms of spectral measures. in addition, for scalar- or operator-valued optimistic sure services that are Laplace transforms, we observe the linked reproducing kernel Hilbert area as an $L^2$-space $L^2(V^*,\mu)$ of vector-valued capabilities and hyperlink the normal translation operators at the reproducing kernel area to multiplication operators on $L^2(V^*,\mu)$, which supplies us subtle info in regards to the norms of those operators.
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