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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

Threshold approximations with corrector for the resolvent of a factorized selfadjoint operator family


Authors: M. Sh. Birman and T. A. Suslina
Translated by: T. A. Suslina
Original publication: Algebra i Analiz, tom 17 (2005), nomer 5.
Journal: St. Petersburg Math. J. 17 (2006), 745-762
MSC (2000): Primary 47A55
Published electronically: July 20, 2006
MathSciNet review: 2241423
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Abstract | References | Similar Articles | Additional Information

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 $ \vert t\vert \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.


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Additional 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

DOI: http://dx.doi.org/10.1090/S1061-0022-06-00927-7
PII: S 1061-0022(06)00927-7
Keywords: Threshold approximations, homogenization, corrector
Received by editor(s): April 11, 2005
Published electronically: July 20, 2006
Additional Notes: Supported by RFBR (grant no. 05-01-01076)
Dedicated: In fond memory of Ol$’$ga Aleksandrovna Ladyzhenskaya
Article copyright: © Copyright 2006 American Mathematical Society