A linear operator between two Banach spaces X and Y is strictly-singular (or Kato)
if it fails to be an isomorphism on any infinite dimensional subspace. A weaker notion
for Banach lattices introduced in [8] is the following one: an operator T from a
Banach lattice X to a Banach space Y is said to be disjointly strictly-singular if there
is no disjoint sequence of non-null vectors (xn)n∈ℕ in X such that the restriction of T
to the subspace [(xn)∞n=1] spanned by the vectors
(xn)n∈ℕ is an isomorphism. Clearly every strictly-singular
operator is disjointly strictly-singular but the converse is not true in general (consider for example the canonic inclusion
Lq[0, 1][rarrhk ]Lp[0, 1] for
1[les ]p<q<∞). In the special case of considering Banach lattices X with a Schauder
basis of disjoint vectors both concepts coincide. The notion of disjointly strictly-singular has turned out to be a useful tool in the study of lattice structure of function
spaces (cf. [7–9]). In general the class of all disjointly strictly-singular operators is not
an operator ideal since it fails to be stable with respect to the composition on the right.
The aim of this paper is to study when the inclusion operators between arbitrary
rearrangement invariant function spaces E[0, 1] ≡ E
on the probability space [0, 1] are disjointly strictly-singular operators.