Flows in Infinite-Dimensional Phase Space Equipped with a Finitely-Additive Invariant Measure
Finitely-additive measures invariant to the action of some groups on a separable infinitedimensional real Hilbert space are constructed. The invariantness of a measure is studied with respect to the group of shifts on a vector of Hilbert space, the orthogonal group and some groups of symplectomorphi...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/5/1161 |
| _version_ | 1797614860302811136 |
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| author | Vsevolod Zh. Sakbaev |
| author_facet | Vsevolod Zh. Sakbaev |
| author_sort | Vsevolod Zh. Sakbaev |
| collection | DOAJ |
| description | Finitely-additive measures invariant to the action of some groups on a separable infinitedimensional real Hilbert space are constructed. The invariantness of a measure is studied with respect to the group of shifts on a vector of Hilbert space, the orthogonal group and some groups of symplectomorphisms of the Hilbert space equipped with the shift-invariant symplectic form. A considered invariant measure is locally finite, σ finite, but it is not countably additive. The analog of the ergodic decomposition of invariant finitely additivemeasures with respect to some groups are obtained. The set of measures that are invariant with respect to a group is parametrized using the obtained decomposition. The paper describes the spaces of complex-valued functions which are quadratically integrable with respect to constructed invariant measures. This space is used to define the Koopman unitary representation of the group of transformations of the Hilbert space. To define the strong continuity subspaces of a Koopman group, we analyze the spectral properties of its generator. |
| first_indexed | 2024-03-11T07:18:14Z |
| format | Article |
| id | doaj.art-1768fc8d056a4c38b069539d93251b3a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T07:18:14Z |
| publishDate | 2023-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-1768fc8d056a4c38b069539d93251b3a2023-11-17T08:09:03ZengMDPI AGMathematics2227-73902023-02-01115116110.3390/math11051161Flows in Infinite-Dimensional Phase Space Equipped with a Finitely-Additive Invariant MeasureVsevolod Zh. Sakbaev0Steklov Institute of Mathematics of Russian Academy of Science, Moscow 119991, RussiaFinitely-additive measures invariant to the action of some groups on a separable infinitedimensional real Hilbert space are constructed. The invariantness of a measure is studied with respect to the group of shifts on a vector of Hilbert space, the orthogonal group and some groups of symplectomorphisms of the Hilbert space equipped with the shift-invariant symplectic form. A considered invariant measure is locally finite, σ finite, but it is not countably additive. The analog of the ergodic decomposition of invariant finitely additivemeasures with respect to some groups are obtained. The set of measures that are invariant with respect to a group is parametrized using the obtained decomposition. The paper describes the spaces of complex-valued functions which are quadratically integrable with respect to constructed invariant measures. This space is used to define the Koopman unitary representation of the group of transformations of the Hilbert space. To define the strong continuity subspaces of a Koopman group, we analyze the spectral properties of its generator.https://www.mdpi.com/2227-7390/11/5/1161A. Weil theoremfinitely-additive measureshift-invariant measure on an infinite-dimensional spaceisometry-invariant measure on a Hilbert spaceKoopman representation of a Hamiltonian flow |
| spellingShingle | Vsevolod Zh. Sakbaev Flows in Infinite-Dimensional Phase Space Equipped with a Finitely-Additive Invariant Measure Mathematics A. Weil theorem finitely-additive measure shift-invariant measure on an infinite-dimensional space isometry-invariant measure on a Hilbert space Koopman representation of a Hamiltonian flow |
| title | Flows in Infinite-Dimensional Phase Space Equipped with a Finitely-Additive Invariant Measure |
| title_full | Flows in Infinite-Dimensional Phase Space Equipped with a Finitely-Additive Invariant Measure |
| title_fullStr | Flows in Infinite-Dimensional Phase Space Equipped with a Finitely-Additive Invariant Measure |
| title_full_unstemmed | Flows in Infinite-Dimensional Phase Space Equipped with a Finitely-Additive Invariant Measure |
| title_short | Flows in Infinite-Dimensional Phase Space Equipped with a Finitely-Additive Invariant Measure |
| title_sort | flows in infinite dimensional phase space equipped with a finitely additive invariant measure |
| topic | A. Weil theorem finitely-additive measure shift-invariant measure on an infinite-dimensional space isometry-invariant measure on a Hilbert space Koopman representation of a Hamiltonian flow |
| url | https://www.mdpi.com/2227-7390/11/5/1161 |
| work_keys_str_mv | AT vsevolodzhsakbaev flowsininfinitedimensionalphasespaceequippedwithafinitelyadditiveinvariantmeasure |