Positivity 12 (2008), 193–208
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020193-16, published online January 11, 2008
Invariant Subspaces of Positive Quasinilpotent
Operators on Ordered Banach Spaces
Hailegebriel E. Gessesse and Vladimir G. Troitsky
Abstract. In this paper we ﬁnd invariant subspaces of certain positive quasi-
nilpotent operators on Krein spaces and, more generally, on ordered Banach
spaces with closed generating cones. In the later case, we use the method of
minimal vectors. We present applications to Sobolev spaces, spaces of differ-
entiable functions, and C*-algebras.
Mathematics Subject Classiﬁcation (2000). Primary: 47A15, Secondary: 46B42,
Keywords. Invariant subspace, minimal vector, ordered normed space, positive
operator, Sobolev space.
0. Introduction and Notations
Lomonosov proved in  that if an operator T on a Banach space is not a multiple
of the identity and commutes with a non-zero compact operator, then there is a
closed (proper non-zero) subspace which is invariant under every operator com-
muting with T . The following result shows that the situation is even better for
positive operators on Banach lattices.
Theorem 0.1 (). Let S and T be two positive commuting operators on a Banach
lattice such that S is quasinilpotent and dominates a non-zero positive compact
operator. Then T has a closed invariant subspace.
Moreover, this invariant subspace can be chosen to be a closed order ideal.
Several variations of this result can be found in [3–5,1,13]. In particular, ST = TS
may be replaced with ST TS. In , the method of minimal vectors was used
to show that the presence of a compact operator can be replaced with a weaker
“localization” condition. In the present paper we extend some of these results
beyond Banach lattices. We show that many of them remain valid in ordered
Banach spaces with closed generating cones. We present applications to spaces of
differentiable functions, C
-algebras, and Sobolev spaces.