Positivity 2: 257–264, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
On the Modulus of C. J. Read’s Operator
Mathematics Department, University of Illinois at Urbana-Champaign, 1409 West Green St.,
Urbana, IL 61801, USA
(Received: 23 February 1998; Accepted: 16 March 1998)
Abstract. Let T :
be the quasinilpotent operator without an invariant subspace constructed
by C. J. Read in . We prove that the modulus of this operator has an invariant subspace (and even
an eigenvector). This answers a question posed by Y. Abramovich, C. Aliprantis and O. Burkinshaw
in [1, 3].
Mathematics Subject Classiﬁcation (1991): 47A15, 47B60, 47B65.
Key words: Banach lattice, invariant subspaces, positive operator.
During the last several years there has been a noticeable increase of interest in
the invariant subspace problem for positive operators on Banach lattices. A rather
complete and comprehensive survey on this topic is presented in , to which
we refer the reader for details and for an extensive bibliography. In particular, the
following theorem was proved in .
THEOREM 1 ([1, 3]). If the modulus of a continuous operator T :
p<∞)exists and is quasinilpotent, then T has a non-trivial closed invariant
subspace which is an ideal.
It follows that each positive quasinilpotent operator on
a nontrivial closed invariant subspace. In the same papers the authors posed the
PROBLEM Does every positive operator on
have an invariant subspace?
Keeping in mind that each operator on
has a modulus and that C. J. Read
in [8, 9, 10] has constructed several operators on
without invariant subspaces,
it was suggested in [1, 3] that the modulus of some of these operators might be
a natural candidate for a counterexample to the above problem. Following this
suggestion, we will be dealing in this paper with the modulus of the quasinilpotent
operator T constructed in . It turns out, quite surprisingly, that not only does
Supported in part by NSF Grant DMS 96-22454.
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