Transformation Semigroups of the Space of Functions that are Square Integrable with Respect to a Translation-Invariant M
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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 V. Zh. Sakbaev
UDC 517.982, 517.983
Abstract. 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 H 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 H 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 H, find invariant subspaces of operators of this semigroup, and obtain conditions of its strong continuity. Keywords and phrases: finitely additive measure, invariant measure on a group, random walk, continuous one-parameter semigroup, generator, resolvent. AMS Subject Classification: 28C20, 81Q05, 47D08
CONTENTS 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Translation Invariant Measures on Banach Spaces . . . . . . . . . . . . . . . . . . . Spaces of Functions That Are Square Integrable with respect to Invariant Measures Semigroups of Translation Operators and Their Properties . . . . . . . . . . . . . . . Measures on Hilbert Spaces Invariant under Translations and Rotations . . . . . . . Random Translations and Their Mathematical Expectations . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
The development of the theory of translation invariant measures on infinite-dimensional vector spaces is complicated by the fact that, due to the Weil theorem (see [16]), the Lebesgue measure on an infinite-dimensional topological vector space does not exist, i.e., there are no nonzero, countablyadditive, σ-finite measures on the σ-ring of Borel subsets of an infinite-dimensional topological vector space, which is invariant under translations by vectors of this space. In this connection, problems on the existence of measures on infinite-dimensional topological vector spaces that are invariant under translations by vectors from a certain maximal admissible subspace (see [15]), on the existence of invariant measures that are not σ-finite (see [3]), and on the existence of measures that are not countably additive (see [10]). For separable
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