Positivity 7: 113–118, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
A counterexample of compact operator between
ZI LI CHEN
Department of Applied Mathematics, Southwest Jiaotong University, Chengdu 610031, P.R. China
Abstract. In this paper we present a counterexample of compact domination compact operator T
such that |T
| is compact but |T | need not compact.
MR Subject Classiﬁcation (1991): 47B65,47B07
Key words: Banach lattice, compact domination, modulus
It is usual to assume when dealing with the lattice properties of operators (operator
spaces) on Banach lattices, that the range spaces of operators are Dedekind com-
plete, or at least σ -Dedekind complete, the reason is to guarantee the modulus of
operator does exist. It has long been a question for the existence of the modulus
of an operator between two Banach lattices. That is, it is hard to determine that
whether or not the modulus of an operator does exist. So it is natural to turn
our attention to the operators with nice behavior, such as compact operators. The
following are the main results of the question for compact operators:
• In 1966, Krengel  constructed a non-regular compact operator, and hence
• In 1992, Abramovich and Wickstead  showed a compact regular operator
• In 1995, Abramovich and Wickstead  gave an example of a compact oper-
ator dominated by a positive compact operator, but its modulus does not exist.
And even if its modulus exists, it need not be compact.
• In 1997, Chen and Wickstead  showed that the linear span of the positive
compact operators need not be complete under the regular norm.
• In 1998, Chen and Wickstead  showed that for each preregular operator T
is regular), the modulus |T
| and the modulus |JT| of the
operator T as into the second dual space, are simultaneous compact or not.
It is a known fact that for all counter-examples the modulus of the compact operat-
ors and the modulus of its conjugate operator are simultaneous compact or not. So
the following question occurs in a natural way.