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- W2078729841 abstract "In this paper W-to-$W^ * $ operator-valued positive definite kernels $K( cdot cdot )$ are defined on $Lambda times Lambda $, where W is a Banach space and $Lambda $ an arbitrary set, and Hibertian varieties $X( cdot )$ with such covariance kernels are studied with the aid of the Kolmogorov–Aronszajn–Pedrick kernel theorem. The general notion of a propagator of $X( cdot )$ is introduced in terms of the action on $Lambda $ of a semi-group $Gamma $, and necessary and sufficient conditions are established for its existence. We show that these conditions simplify substantially when the semi-group $Gamma $ is involutory, especially so when $Lambda = Gamma $. For $Gamma = Lambda $ equal to a unitized Banach algebra with isometric involution, these conditions subsume those given by Stinespring for $C^ * $-algebras. If for Hilbert spaces W and semi-groups $Lambda $, dilations are redefined in terms of isometrics rather than projections, then the dilation $tilde R( cdot )$ of a given W-to-W operator-valued function $R( cdot )$ on $Lambda $ is precisely the propagator of a Hilbertian variety whose covariance kernel $K( cdot cdot )$ satisfies $K( cdot ,0) = R( cdot )$. Dilation theorems are thus rendered explicit, and their method of proof routinized. From our results on propagators we deduce a simplified version of Nagy’s principal theorem in which his translational inequality is mitigated, and the Bram version of Halmos’s theorem on subnormal operators. Dilation theorems such as those of Lebow, Arveson and Naimark are shown to fit into this approach." @default.
- W2078729841 created "2016-06-24" @default.
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- W2078729841 date "1978-06-01" @default.
- W2078729841 modified "2023-09-24" @default.
- W2078729841 title "Dilations as Propagators of Hilbertian Varieties" @default.
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- W2078729841 doi "https://doi.org/10.1137/0509027" @default.
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