Positivity 1: 103–124, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Positivity and Stability for One-Sided Coupled
(Accepted: 4 March 1996)
Abstract. Many evolutionary systems can be described by an abstract Cauchy problem governed by
an operator matrix. Assuming this problem to be “one-sided coupled" and “well-posed" we study in
this paper the positivity and the stability of the associated matrix semigroup. The abstract results are
illustrated by several examples.
Mathematics Subject Classiﬁcations (1991): 47D06.
Keywords: abstract Cauchy problem, operator matrices, positive semigroups
Our starting point in this paper is the abstract Cauchy problem
is given by an operator matrix on the product
= E F
of two Banach
. While in Nagel (1989), Engel (1996, Engel and Hengstberger
(1996) one can ﬁnd a detailed analysis of the “well-posedness" of (ACP) we are
here interested in the qualitative behaviour of its solutions.
To this end we assume that (ACP) is well-posed, i.e., that
generates a strongly
. We then ask under which conditions for each
positive initial value
3 t 7! U(t)=T(t)X
assumes only positive values. Or, to state it in semigroup terms, when does
generate a positive semigroup
In order to give sense to this question we require the spaces
Banach lattices and refer to Schaefer (1974) or Aliprantis and Burkinshaw (1985)
for a comprehensive treatment of these spaces. Then the product space
= E F
becomes again a Banach lattice with respect to the coordinate-wise order and, e.g.,
+ ky k
p 2 [
fkxk; ky kg
Firstproof, pips: 136215 MATHKAP
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