Well-posedness of the cauchy problem for complete second-order operator-differential equations
For the equation y″(t)+Ay′(t)+By(t)=0, where A and B are arbitrary commuting normal operators in a Hilbert space H, we obtain a necessary and sufficient condition for well-posedness of the Cauchy problem in the space of initial data D(B)×(D(A)∩D(|B|1/2)) and for weak well-posedness of the Cauchy pro...
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| Datum: | 1996 |
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
1996
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| author | Shklyar, A. Ya. Шкляр, А. Я. Шкляр, А. Я. |
| author_facet | Shklyar, A. Ya. Шкляр, А. Я. Шкляр, А. Я. |
| author_sort | Shklyar, A. Ya. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:29:03Z |
| description | For the equation y″(t)+Ay′(t)+By(t)=0, where A and B are arbitrary commuting normal operators in a Hilbert space H, we obtain a necessary and sufficient condition for well-posedness of the Cauchy problem in the space of initial data D(B)×(D(A)∩D(|B|1/2)) and for weak well-posedness of the Cauchy problem in H×H_(|A|+|B|1/2+1). This condition is expressed in terms of location of the joint spectrum of the operators A and B in C 2. In terms of location of the spectrum of the operator pencil z 2+Az+B in C 1, such a condition cannot be written. |
| first_indexed | 2026-03-24T03:13:39Z |
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| id | umjimathkievua-article-5272 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | rus English |
| last_indexed | 2026-03-24T03:13:39Z |
| publishDate | 1996 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ae/728b74a30e63808e16806444acc3bdae.pdf |
| spelling | umjimathkievua-article-52722020-03-18T21:29:03Z Well-posedness of the cauchy problem for complete second-order operator-differential equations Корректность задачи Коші для полных дифференциально-операторных уравнений второго порядка Shklyar, A. Ya. Шкляр, А. Я. Шкляр, А. Я. For the equation y″(t)+Ay′(t)+By(t)=0, where A and B are arbitrary commuting normal operators in a Hilbert space H, we obtain a necessary and sufficient condition for well-posedness of the Cauchy problem in the space of initial data D(B)×(D(A)∩D(|B|1/2)) and for weak well-posedness of the Cauchy problem in H×H_(|A|+|B|1/2+1). This condition is expressed in terms of location of the joint spectrum of the operators A and B in C 2. In terms of location of the spectrum of the operator pencil z 2+Az+B in C 1, such a condition cannot be written. Для рівняння $y″(t) + Ay′(t) + By(t) = 0$, де $А$ і $В$ —довільні нормальні оператори в гільбертовому просторі $H$, що комутують, дано необхідну і достатню умову для коректності задачі Коші в просторі початкових даних $D(B) × (D(A) ∩ D(|B|^{1/2)})$ і для слабкої коректності задачі Коші в Я $H × H_(|A| + |B|^{1/2} + 1)$. Цю умову виражено в термінах розміщення в $C^2$ сумісного спектра операторів $А$ і $В$. В термінах розміщення в $С^1$ спектра операторного пучка $z^2 + Az + B $ таку умову записати неможливо. Institute of Mathematics, NAS of Ukraine 1996-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5272 Ukrains’kyi Matematychnyi Zhurnal; Vol. 48 No. 7 (1996); 999-1006 Український математичний журнал; Том 48 № 7 (1996); 999-1006 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5272/7236 https://umj.imath.kiev.ua/index.php/umj/article/view/5272/7237 Copyright (c) 1996 Shklyar A. Ya. |
| spellingShingle | Shklyar, A. Ya. Шкляр, А. Я. Шкляр, А. Я. Well-posedness of the cauchy problem for complete second-order operator-differential equations |
| title | Well-posedness of the cauchy problem for complete second-order operator-differential equations |
| title_alt | Корректность задачи Коші для полных дифференциально-операторных
уравнений второго порядка |
| title_full | Well-posedness of the cauchy problem for complete second-order operator-differential equations |
| title_fullStr | Well-posedness of the cauchy problem for complete second-order operator-differential equations |
| title_full_unstemmed | Well-posedness of the cauchy problem for complete second-order operator-differential equations |
| title_short | Well-posedness of the cauchy problem for complete second-order operator-differential equations |
| title_sort | well-posedness of the cauchy problem for complete second-order operator-differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5272 |
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