Positivity (2005) 9:273–286 © Springer 2005
Some Open Problems and Conjectures Associated
with the Invariant Subspace Problem
Y. A. ABRAMOVICH
, C. D. ALIPRANTIS
, G. SIROTKIN
V. G. TROITSKY
Department of Economics, Krannert School of Management, Purdue University, 403 West
State Street, W. Lafayette, IN 47907-2056, USA. (E-mail: firstname.lastname@example.org);
Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA.
Department of Mathematics, University of Alberta, Edmonton, AB Canada T6G 2G1.
Received 2 August 2004; accepted 8 September 2004
Abstract. There is a subtle difference as far as the invariant subspace problem is con-
cerned for operators acting on real Banach spaces and operators acting on complex
Banach spaces. For instance, the classical hyperinvariant subspace theorem of Lomonosov
[Funktsional. Anal. Anal. i Prilozhen 7(3)(1973), 55–56. (Russian)], while true for complex
Banach spaces is false for real Banach spaces. When one starts with a bounded operator
on a real Banach space and then considers some “complexiﬁcation technique” to extend
the operator to a complex Banach space, there seems to be no pattern that indicates any
connection between the invariant subspaces of the “real” operator and those of its “com-
plexiﬁcations.” The purpose of this note is to examine two complexiﬁcation methods of
an operator T acting on a real Banach space and present some questions regarding the
invariant subspaces of T and those of its complexiﬁcations.
Mathematics Subject Classiﬁcation 1991: 47A15, 47C05, 47L20, 46B99
Key words: invariant subspaces, complexiﬁcation, algebra of operators
For unexplained terminology in this paper, we refer the reader to . If
Y is an arbitrary (real or complex) Banach space, then L(Y ) will denote
the algebra of all bounded operators on Y .IfT ∈ L(Y ), then Lat(T )
will denote the collection of all closed T -invariant subspaces of Y . Like-
wise if A is an algebra of bounded operators on Y , i.e., a subalgebra of
L(Y ), then Lat A is the collection of all closed subspaces of Y that are
The research of Aliprantis is supported by the NSF Grants EIA-0075506, SES-0128039 and
DMI-0122214 and the DOD Grant ACI-0325846.