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Type: Journal article
Title: Cosine families and semigroups really differ
Author: Bobrowski, A.
Chojnacki, W.
Citation: Journal of Evolution Equations, 2013; 13(4):897-916
Publisher: Birkhaeuser Science
Issue Date: 2013
ISSN: 1424-3199
Statement of
Adam Bobrowski and Wojciech Chojnacki
Abstract: We reveal three surprising properties of cosine families, distinguishing them from semigroups of operators: (1) A single trajectory of a cosine family is either strongly continuous or not measurable. (2) Pointwise convergence of a sequence of equibounded cosine families implies that the convergence is almost uniform for time in the entire real line; in particular, cosine families cannot be perturbed in a singular way. (3) A non-constant trajectory of a bounded cosine family does not have a limit at infinity; in particular, the rich theory of asymptotic behaviour of semigroups does not have a counterpart for cosine families. In addition, we show that equibounded cosine families that converge strongly and almost uniformly in time may fail to converge uniformly.
Keywords: Primary 47D09
Secondary 60J65
Cosine family
semigroup of operators
strong measurability
strong continuity
Rights: © 2013 The Author(s). Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
DOI: 10.1007/s00028-013-0208-0
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