Geodesic
Gaussian semigroups of operators in the space of Borel functions on a separable Hilbert space
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1320-1340

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The concept of a Gaussian family of Borel measures on a separable Hilbert space is introduced in the paper. Necessary and sufficient conditions are found under which a Gaussian family of measures generates a semigroup of operators on the space of complex bounded Borel functions. These conditions are expressed in the form of a system of functional equations and initial conditions for operator-valued functions on the real semi-axis. A system of differential equations is derived from the system of functional equations and it is proved that the Cauchy problem has a unique solution for it. Several examples of Gaussian semigroups of operators are given.
Keywords: gaussian semigroup of operators, Gaussian family of Borel measures, operator Riccati differential equation, determinant of infinite order, system of functional equations. 
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     author = {O. E. Galkin and S. Yu. Galkina and I. Yu. Yastrebova},
     title = {Gaussian semigroups of operators in the space of {Borel} functions on a separable {Hilbert} space},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
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     year = {2023},
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     url = {http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a66/}
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O. E. Galkin; S. Yu. Galkina; I. Yu. Yastrebova. Gaussian semigroups of operators in the space of Borel functions on a separable Hilbert space. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1320-1340. http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a66/