Positivity 13 (2009), 299–306
2009 Birkh¨auser Verlag Basel/Switzerland
One of the most fruitful activities that takes place at a conference is the
interchange of problems. Often, in later years the success of a conference may be
judged by the number of problems raised there which are eventually solved. The
papers in the remainder of this volume contain several problems, but here are some
more that did not ﬁt naturally into any of those papers. I hope to see solutions of
at least some of these problems in the pages of Positivity in years to come.
1. Some questions about operators on Banach lattices
Z. L. Chen
Department of Mathematics,
Southwest Jiaotong University,
Chengdu 610031, P.R. China
Let E and F be Banach lattices, T,K ∈L(E,F) with 0 ≤ T ≤ K.We
have shown that if either E
or F has an order continuous norm, T is an almost
interval-preserving or lattice homomorphism and K is compact then T is compact.
Similarly if E and F are Banach lattices, T,U ∈L(E,F) with 0 ≤ T ≤ U
with U being weakly compact and T is either almost interval-preserving or a lattice
homomorphism, then T is weakly compact.
Problem 1.1. Is a similar result true for Dunford-Pettis operators? To be pre-
cise, if E and F are Banach lattices, T,U ∈L(E,F) with 0 ≤ T ≤ U, U is
a Dunford-Pettis operator and T is either almost interval-preserving or a lattice
homomorphism, then is T necessarily a Dunford-Pettis operator?
We would like to ask the following two concrete questions concerning the
density of regular operators in continuous operators.
Problem 1.2. For 1 <p<q<∞,isL
) dense in (L(
Problem 1.3. For 1 <p<q<∞,isL
[0, 1]) dense in (L(L
[0, 1], ·)?