Optimal properties of contraction semigroups in banach spaces |
| |
Authors: | Jean Bernard Baillon Sylvie Guerre-Delabrière |
| |
Affiliation: | (1) Intitut de Maths et d'Info, Université de Lyon I, Bat 101, 18 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France;(2) Equipe d'Analyse, Boite 186, Université Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, France |
| |
Abstract: | Suppose that(T t )t>0 is aC 0 semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ −1(Jx)={x}, whereJ is the duality mapping fromX toP(X *). If |<T t x,f>|→1, whent→+∞ for somef∈X *, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible. If the hypothesisJ −1(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ −1(Jx) is weakly compact, then if |<T t x, f>|→1, whent→+∞ for somef∈X *, ‖f‖≤1, there existsy∈J −1(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ1 orL 1. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|