Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations
We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on () ( a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2022
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211717 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations. Michael Skeide. SIGMA 18 (2022), 071, 8 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862533148831645696 |
|---|---|
| author | Skeide, Michael |
| author_facet | Skeide, Michael |
| citation_txt | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations. Michael Skeide. SIGMA 18 (2022), 071, 8 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on () ( a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup). The proof is by general abstract nonsense (taken from Arveson's classification of ₀-semigroups on () by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy. All other results that make similar statements (especially, [Mem. Amer. Math. Soc. 240 (2016), vi+126 pages, arXiv:0901.1798]) for more general *-algebras have been proved later by suitable adaptations of the methods exposed here. (They use Hilbert module techniques, which we carefully avoid here to make the result available without any appeal to Hilbert modules.)
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| first_indexed | 2026-03-12T14:20:35Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211717 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T14:20:35Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Skeide, Michael 2026-01-09T12:43:12Z 2022 Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations. Michael Skeide. SIGMA 18 (2022), 071, 8 pages 1815-0659 2020 Mathematics Subject Classification: 46L55; 46L53; 81S22; 60J25 arXiv:0809.3538 https://nasplib.isofts.kiev.ua/handle/123456789/211717 https://doi.org/10.3842/SIGMA.2022.071 We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on () ( a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup). The proof is by general abstract nonsense (taken from Arveson's classification of ₀-semigroups on () by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy. All other results that make similar statements (especially, [Mem. Amer. Math. Soc. 240 (2016), vi+126 pages, arXiv:0901.1798]) for more general *-algebras have been proved later by suitable adaptations of the methods exposed here. (They use Hilbert module techniques, which we carefully avoid here to make the result available without any appeal to Hilbert modules.) I would like to thank Rajarama Bhat, Malte Gerhold, Martin Lindsay, and Orr Shalit for several valuable comments and bibliographical hints, which improved the final version in several ways. In particular, I wish to express my gratitude to the referees. One of them spotted a seriously unclear point in a definition and a small gap in an argument. This work was supported by research funds of the University of Molise and the Italian MIUR under PRIN 2007. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations Article published earlier |
| spellingShingle | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations Skeide, Michael |
| title | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations |
| title_full | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations |
| title_fullStr | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations |
| title_full_unstemmed | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations |
| title_short | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations |
| title_sort | spatial markov semigroups admit hudson-parthasarathy dilations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211717 |
| work_keys_str_mv | AT skeidemichael spatialmarkovsemigroupsadmithudsonparthasarathydilations |