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Transformation Semigroups of the Space of Functions that are Square Integrable with Respect to a Translation-Invariant Measure on a Banach Space

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We examine measures on a Banach space E that are invariant under shifts by arbitrary vectors of the space and are additive extensions of a set function defined on the family of bars with converging products of edge lengths that do not satisfy the σ-finiteness condition and, perhaps, the countable additivity condition. We introduce the Hilbert space of complex-valued functions of the space E of functions that are square integrable with respect to a shift-invariant measure. We analyze properties of semigroups of shift operators in the space and the corresponding generators and resolvents. We obtain a criterion of the strong continuity of such semigroups. We introduce and examine mathematical expectations of operators of shifts along random vectors by a one-parameter family of Gaussian measures that form a semigroup with respect to the convolution. We prove that the family of mathematical expectations is a one-parameter semigroup of linear self-adjoint contraction mappings of the space , find invariant subspaces of operators of this semigroup, and obtain conditions of its strong continuity.

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Correspondence to V. Zh. Sakbaev.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 151, Quantum Probability, 2018.

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Sakbaev, V.Z. Transformation Semigroups of the Space of Functions that are Square Integrable with Respect to a Translation-Invariant Measure on a Banach Space. J Math Sci 252, 72–89 (2021).

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