Abstract and Applied Analysis
Volume 2008 (2008), Article ID 651294, 11 pages
doi:10.1155/2008/651294
Research Article
On the Adjoint of a Strongly Continuous Semigroup
1Universidad de los Andes, Mérida, Venezuela
2Universidad Simón Bolivar, Caracas, Venezuela
Received 1 December 2006; Revised 14 November 2007; Accepted 9 December 2007
Academic Editor: Simeon Reich
Copyright © 2008 Diómedes Bárcenas and Luis Gerardo Mármol. 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.
Abstract
Using some techniques from vector integration,
we prove the weak measurability of the adjoint of strongly
continuous semigroups which factor through Banach spaces
without isomorphic copy of
l1;
we also prove the strong continuity away from zero of the
adjoint if the semigroup factors through Grothendieck spaces.
These results are used, in particular, to characterize the
space of strong continuity of
{T**(t)}t≥0,
which, in addition, is also characterized for abstract
L- and M-spaces. As a
corollary, it is proven that abstract
L-spaces with no copy of
l1
are finite-dimensional.