Positivity 1: 171–180, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Multiplication and Compact-friendly Operators
Dedicated to the memory of our friend C. B. Huijsmans (1946–1997)
Y. A. ABRAMOVICH, C. D. ALIPRANTIS and O. BURKINSHAW
Department of Mathematical Sciences, IUPUI, Indianapolis, IN 46202-3216, USA
(Received: 12 November 1996; accepted: 12 May 1997)
Abstract. During the last few years the authors have studied extensively the invariant subspace
problem of positive operators; see  for a survey of this investigation. In  the authors introduced
the class of compact-friendly operators and proved for them a general theorem on the existence of
invariant subspaces. It was then asked if every positive operator is compact-friendly. In this note, we
present an example of a positive operator which is not compact-friendly but which, nevertheless, has
a non-trivial closed invariant subspace.
In the process of presenting this example, we also characterize the multiplication operators that
commute with non-zero ﬁnite-rank operators. We show, among other things, that a multiplication
commutes with a non-zero ﬁnite-rank operator if and only the multiplier function
constant on some non-empty open set.
Mathematics Subject Classiﬁcations (1991): 47BE81, 47B60, 47B65
Keywords: Banach lattice, positive operator, multiplication operator, compact-friendly operator,
commutant, invariant subspaces
We gather here a few basic notions needed for our discussion. All Banach spaces
encountered in this work are assumed to be real. For details about Banach lattices
and positive operators, we refer the reader to .
The word “operator” will be synonymous with “linear operator.” An operator
on a Banach lattice
is positive if
0 for each
0. Recall that a positive
dominates an operator
x 2 E
. Since positive
operators between Banach lattices are continuous, each operator dominated by a
positive operator is automatically continuous.
The compact-friendly operators were introduced in  as follows.
DEFINITION 1.1. A positive operator
E ! E
is said to be compact-friendly
if there exist three non-zero operators
R; K; A
E ! E
compact such that
RB = BR; R
icpc Firstproof, pips: 141204 MATHKAP
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