Positivity 7: 73–80, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
On positive strictly singular operators and
and FRANCISCO L. HERNÁNDEZ
Area de Física y Matemáticas Aplicadas. Escet. Universidad Rey Juan Carlos 28933 Madrid,
Spain. E-mail: jﬂores@escet.urjc.es;
Departamento de Análisis Matemático. Facultad de
Matemáticas. Universidad Complutense 28040 Madrid, Spain. E-mail: email@example.com
Abstract. We study the domination problem by positive strictly singular % operators between
Banach lattices. Precisely we show that if E and %F are two Banach lattices such that the norms
and F are %order continuous and E satisﬁes the subsequence splitting property, %and %0 ≤
S ≤ T : E → F are two positive operators, then T strictly %singular implies S strictly singular. The
special case of %endomorphisms is also considered. Applications to the class of %strictly co-singular
(or Pelczynski) operators are given too.
Mathematics Subject Classiﬁcation 2000: 47B65
The problem of domination by positive compact operators between Banach lattices
was solved by Dodds and Fremlin : given a Banach lattice E with order continu-
ous dual norm, an order continuous Banach lattice F and two positive operators
0 ≤ S ≤ T : E → F, the operator S is compact if T is so. A similar problem
has been considered in the class of weakly compact operators by Abramovich 
and in a general form by Wickstead . On the other hand Kalton and Saab
 have proved that the operator S is Dunford–Pettis if T is so, provided the
Banach lattice F is order continuous, giving thus an answer to a question posed by
Aliprantis and Burkinshaw. More recently Wickstead  has studied converses
for the Dodds–Fremlin and Kalton–Saab theorems.
In this note we present several results concerning the domination by positive
strictly singular operators obtained recently in . Recall that a bounded operator
T between two Banach spaces X and Y issaidtobestrictly singular (or Kato) if
the restriction of T to any inﬁnite-dimensional (closed) subspace of X is not an
isomorphism. The class of all strictly singular operators is a closed operator ideal
(in the sense of ), which contains the ideal of all compact operators and is not
stable by duality (). A well-known characterization of strict singularity is the
following: an operator T : X → Y is strictly singular if and only if every inﬁnite-
dimensional subspace M of X contains an inﬁnite dimensional subspace N of M
such that the restriction of T to N is compact (cf. , ). The class of all strictly
singular operators between X and Y will be denoted by SS(X, Y ).
We remark that in general the problem of domination in the class of strictly
singular operators has a negative answer (Examples 0.7 and 0.8) and we present
some natural sufﬁcient conditions on the Banach lattices E and F which yield