Threshold approximations with corrector for the resolvent of a factorized selfadjoint operator family
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M. Sh. Birman and T. A. Suslina
Translated by: T. A. Suslina - St. Petersburg Math. J. 17 (2006), 745-762
- DOI: https://doi.org/10.1090/S1061-0022-06-00927-7
- Published electronically: July 20, 2006
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Abstract:
In a Hilbert space, a family of operators admitting a factorization $A(t)= X(t)^*X(t)$, where $X(t)=X_0 +tX_1$, $t \in \mathbb R$, is considered. It is assumed that the subspace $\mathfrak {N} = \operatorname {Ker} A(0)$ is finite-dimensional. For the resolvent $(A(t)+\varepsilon ^2 I)^{-1}$ with small $\varepsilon$, an approximation in the operator norm is obtained on a fixed interval $|t| \le t^0$. This approximation involves the so-called “corrector”; the remainder term is of order $O(1)$. The results are aimed at applications to homogenization of periodic differential operators in the small period limit. The paper develops and refines the results of Chapter 1 of our paper in St. Petersburg Math. J. 15 (2004), 639–714.References
- M. Sh. Birman and T. A. Suslina, Periodic second-order differential operators. Threshold properties and averaging, Algebra i Analiz 15 (2003), no. 5, 1–108 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 5, 639–714. MR 2068790, DOI 10.1090/S1061-0022-04-00827-1
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
Bibliographic Information
- M. Sh. Birman
- Affiliation: Department of Physics, St. Petersburg State University, Ulyanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
- Email: mbirman@list.ru
- T. A. Suslina
- Affiliation: Department of Physics, St. Petersburg State University, Ulyanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
- Email: suslina@list.ru
- Received by editor(s): April 11, 2005
- Published electronically: July 20, 2006
- Additional Notes: Supported by RFBR (grant no. 05-01-01076)
- © Copyright 2006 American Mathematical Society
- Journal: St. Petersburg Math. J. 17 (2006), 745-762
- MSC (2000): Primary 47A55
- DOI: https://doi.org/10.1090/S1061-0022-06-00927-7
- MathSciNet review: 2241423
Dedicated: In fond memory of Ol$’$ga Aleksandrovna Ladyzhenskaya