Positivity 8: 123–126, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Invariant Subspaces for Positive Operators Acting
on a BanachSpace withMarkushevichBasis
Z. ERCAN and S. ONAL
Middle East Technical University, Department of Mathematics 06531 Ankara, Turkey. E-mail:
(Received 6 May 2001; accepted 22 September 2002)
Abstract. We introduce ‘weak quasinilpotence’ for operators. Then, by substituting ‘Markushevich
basis’ and ‘weak quasinilpotence at a nonzero vector’ for ‘Schauder basis’ and ‘quasinilpotence at a
nonzero vector’, respectively, we answer a question on the invariant subspaces of positive operators
Key words: Markushevich basis, positive operators, invariant subspace
1 Weakly quasinilpotent operators.
Throughout this paper we suppose vector spaces are nonzero. A vector space X
is called an ordered vector space under the partial order if, for each z ∈X,
0 ∈R, x +z y +z and x y whenever x y in X. An operator T
on a ordered vector space X is called positive if Tx0 for all x 0. If X is a
Banachspace witha (Schauder) basis x
, then X is an ordered vector space under
the order ,0x =
for each n. In , a bounded operator T
on a Banachspace X is said to be quasinilpotent at x
the following theorem was proved in .
THEOREM 1. Let T be an operator on a Banach space X with a Schauder basis
. Suppose that T is positive. If T commutes with a nonzero positive operator
that is quasinilpotent at 0 <x
then T has a nontrivial closed invariant subspace.
To generalize the above theorem we introduce the following:
DEFINITION 1. We say that an operator T on a Hausdorff topological vector
space X is weakly quasinilpotent at x
for each f ∈X
, where X
denotes the topological dual of X.
Mathematics Subject Classiﬁcation (2000). Primary: 47B65, 46B15, Secondary: 47A15