Positivity 4: 119–130, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
On the Peripheral Spectrum of Order Continuous,
Department of Mathematics and statistics, Texas Tech University, Lubbock, Texas 79409, USA
(Received 3 October 1996; accepted 16 April 1999)
Abstract. Let E be an order complete Banach function lattice and T a positive, eventually compact,
order continuous operator on E. We study necessary conditions under which the peripheral spectrum
of T is fully cyclic in terms of certain bands of the underlying Banach function lattice E.Asetof
sufﬁcient conditions is also given. Examples are presented to demonstrate our methods.
Mathematics Subject Classiﬁcation 1991: 47B65
Key words: Order continuous, Perron-Frobenius theory, Principal T-band
The classical Perron-Frobenius theory concerning the distribution of eigenvalues of
nonnegative matrices has been extended in various ways to the setting of positive
operators. Jang and Victory  describe the positivity structure for the algebraic
eigenspace belonging to the spectral radius for positive, eventually compact op-
erators on Banach lattices with order continuous norm. The idea they used was
motivated by the graph-theoretic techniques in the study of matrices  and the
derived positivity structure of the algebraic eigenspace was later used to study the
nonnegative solvability of certain linear operator equations .
In a more recent study, Grobler and Reinecke  consider eventually compact
and order continuous positive operators on order complete Banach lattices. By
using the order continuity and eventual compactness of positive operators, they
provide a Zaanen basis  for the algebraic eigenspace belonging to the spectral
Our main objective in this work is to study conditions under which a posit-
ive, eventually compact, order continuous operator has a fully cyclic peripheral
spectrum. Our conditions are based on the analysis of the structure of algebraic
eigenspaces belonging to the peripheral spectrum, which can be viewed as a gener-
alization of results in [3, 4]. In particular, a class of positive operators having fully
cyclic peripheral spectrum is shown using our main results. The results we obtain
here in which accessibility between certain bands in the underlying Banach func-