Abstract
It is shown that if A generates a bounded cosine operator function on a UMD space X, then i(−A)1/2 generates a bounded C 0-group. The proof uses a transference principle for cosine functions.
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Acknowledgments
This article was completed while the author enjoyed a two-month stay at the Institute of Mathematics of the Polish Academy of Sciences in Warsaw and at the Nikolaus-Copernicus University in Torun, Poland. The stay was supported by the EU Marie Curie “Transfer of Knowledge” project “Operator Theory Methods for Differential Equations” (MTKD-CT-2005-030042, TODEQ), and the author is grateful for this support and the kind invitation by J. Zemánek (Warsaw) and Y. Tomilov (Torun). The author is also grateful to Bernhard Haak (Bordeaux) who made valuable comments on a previous version of this article.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Haase, M. The group reduction for bounded cosine functions on UMD spaces. Math. Z. 262, 281–299 (2009). https://doi.org/10.1007/s00209-008-0373-y
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DOI: https://doi.org/10.1007/s00209-008-0373-y
Keywords
- Cosine function
- Transference principle
- C 0-semigroup
- Group
- Functional calculus
- UMD space
- Fattorini’s theorem