Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timişoara, C. Coposu Boulevard. no. 4, 300223 Timişoara, Romania
Copyright © 2010 Adina Luminiţa Sasu. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We provide a complete diagram of the relation
between the admissibility of pairs of Banach function spaces and
the exponential dichotomy of evolution families on the
real line. We prove that if and are two Banach function spaces with the
property that either or , then the admissibility of the pair
implies the existence of the exponential dichotomy. We
study when the converse implication holds and show that the
hypotheses on the underlying function spaces cannot be dropped and
that the obtained results are the most general in this topic.
Finally, our results are applied to the study of exponential
dichotomy of -semigroups.