This is an announcement for the paper "A version of Lomonosov's theorem for collections of positive operators" by Alexey I. Popov and Vladimir G. Troitsky.
Abstract: It is known that for every Banach space X and every proper WOT-closed subalgebra A of L(X), if A contains a compact operator then it is not transitive. That is, there exist non-zero x in X and f in X* such that f(Tx)=0 for all T in A. In the case of algebras of adjoint operators on a dual Banach space, V.Lomonosov extended this as follows: without having a compact operator in the algebra, |f(Tx)| is less than or equal to the essential norm of the pre-adjoint operator T_* for all T in A. In this paper, we prove a similar extension (in case of adjoint operators) of a result of R.Drnovsek. Namely, we prove that if C is a collection of positive adjoint operators on a Banach lattice X satisfying certain conditions, then there exist non-zero positive x in X and f in X* such that f(Tx) is less than or equal to the essential norm of T_* for all T in C.
Archive classification: math.FA math.OA
Mathematics Subject Classification: 47B65; 47A15
The source file(s), lom-drnov.tex: 31715 bytes, is(are) stored in gzipped form as 0807.3327.gz with size 10kb. The corresponding postcript file has gzipped size 86kb.
Submitted from: vtroitsky@math.ualberta.ca
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/0807.3327
or
http://arXiv.org/abs/0807.3327
or by email in unzipped form by transmitting an empty message with subject line
uget 0807.3327
or in gzipped form by using subject line
get 0807.3327
to: math@arXiv.org.