Positivity 8: 109–122, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Invariant Subspaces for Compact-Friendly
Operators in Sobolev Spaces
MARIA CRISTINA ISIDORI and ANNA MARTELLOTTI
Departmant of Mathematics, University of Perugia, Via Vanvitelli, 1-06123 Perugia, Italy;
E-mail: email@example.com / firstname.lastname@example.org
(Received and accepted 10 March 2003)
Abstract. In this note we extend the concept of compact-friendlyness, deﬁned in the literature for
operators on a Banach lattice, to the case of operators on Sobolev spaces and derive the existence of
invariant subspaces for compact-friendlyoperators of this type.
Mathematics Subject Classiﬁcations (1991): 47B65, 47A15
Key words: Sobolev spaces, compact-friendlyoperator,invariantsubspaces, multiplication operators
The search for non trivial invariant subspaces of linear and continuous operators
has been a fascinating challenge for the mathematicians since the verybeginning
of Operator Theory.
RecentlyAbramovich et al. in  established existence theorems of invariant
subspaces for positive operators in Banach lattices.
Again in  theyintroduced a new class of positive operators, the compact-
DEFINITION 1.1. A positive operator B on a Banach lattice E dominates an
operator T provided Bx Tx, for each x ∈E. B is compact-friendly if
there exist three non zero operators R, K, A: E →E with R and K compact such
(i) RB =BR;
(ii) R dominates A;
(iii) K dominates A.
Remember that an operator BX→X on a Banach space X is quasinilpotent in
∈X if lim
In  the authors show that everycompact-friendlyoperator which is quasinil-
potent in a positive vector has a closed invariant non trivial subspace.
This paper is dedicated to the memoryof our dear and unforgettable friend Yuri Abramovich.