一个C_0-半群的生长性（以其余生成元为特征）
The growth of a C_0-semigroup characterised by its cogenerator
摘要：
本文讨论了巴拿赫空间上C_0-半群的压缩性、有界性和多项式有界性。特别地，我们推广了Gomilko和Brenner，Thomee的结果，证明了半群的多项式有界性蕴涵了它的余生成元的多项式有界性.算例表明，该方法是最优的。对于解析半群，我们证明了逆命题成立，即，余生成元的多项式有界性蕴涵着半群的多项式有界性。此外，我们还通过（C^2，\| \cdot \|_p），p \neq 2中的简单例子证明了我们关于收缩性的刻画的结果是精确的.这些例子也表明，著名的Foias-Sz.-关于希尔伯特空间上压缩半群的余生成元的Nagy定理在（C^2，\| \cdot \|_p）（p \neq 2）中失效.
英文摘要：
We characterise contractivity, boundedness and polynomial boundedness for a C_0-semigroup on a Banach space in terms of its cogenerator V (or the Cayley transform of the generator) or its resolvent. In particular, we extend results of Gomilko and Brenner, Thomee and show that polynomial boundedness of a semigroup implies polynomial boundedness of its cogenerator. As is shown by an example, the result is optimal. For analytic semigroups we show that the converse holds, i.e., polynomial boundedness of the cogenerators implies polynomial boundedness of the semigroup. In addition, we show by simple examples in (C^2,\|\cdot\|_p), p \neq 2, that our results on the characterisation of contractivity are sharp. These examples also show that the famous Foias-Sz.-Nagy theorem on cogenerators of contraction semigroups on Hilbert spaces fails in (C^2,\|\cdot\|_p) for p\neq 2.
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