Zero-two law for cosine families

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• 作者：Felix L. Schwenninger ; Hans Zwart
• 关键词：Primary 47D09 ; Secondary 47D06 ; Cosine families ; Semigroup of operators ; Zero ; two law ; Zero ; one law
• 刊名：Journal of Evolution Equations
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：15
• 期：3
• 页码：559-569
• 全文大小：423 KB
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  3.Bobrowski A., Chojnacki W.: Isolated points of the set of bounded cosine families, bounded semigroups, and bounded groups on a Banach space. Studia Mathematica, 217(3), 219鈥?41 (2013)MATH MathSciNet CrossRef
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• 作者单位：Felix L. Schwenninger (1)
  Hans Zwart (1)
  
  1. Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1424-3202

文摘

For \({\left(C(t)\right)_{t \geq 0}}\) being a strongly continuous cosine family on a Banach space, we show that the estimate \({\limsup_{t \to 0^{+}} \|C(t) - I\| < 2}\) implies that C(t) converges to I in the operator norm. This implication has become known as the zero-two law. We further prove that the stronger assumption of \({\sup_{t \geq 0} \|C(t) - I\| < 2}\) yields that C(t) = I for all \({t \geq 0}\). For discrete cosine families, the assumption \({\sup_{n \in \mathbb{N}} \|C(n) - I\| \leq r < \frac{3}{2}}\) yields that C(n) = I for all \({n \in \mathbb{N}}\). For \({r \geq \frac{3}{2}}\), this assertion does no longer hold. Mathematics Subject Classification Primary 47D09 Secondary 47D06