Positivity 4: 397–402, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
Embeddings of Rearrangement Invariant Spaces
that are not Strictly Singular
S. J. MONTGOMERY-SMITH
and E. M. SEMENOV
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.
Department of Mathematics, Voronezh State University,
Universitetskaya pl.1, Voronezh 394693, Russia. E-mail: email@example.com
(Received 23 December 1998; accepted 7 November 1999)
Abstract. We give partial answers to the following conjecture: the natural embedding of a re-
arrangement invariant space E into L
([0, 1]) is strictly singular if and only if G does not embed
into E continuously, where G is the closure of the simple functions in the Orlicz space L
(x) = exp(x
) − 1.
A.M.S. Classiﬁcation (1991): Primary 46E30, 47B38; Secondary 60G50
Key words: rearrangement invariant space, strictly singular mapping, Rademacher function, Orlicz
In this paper we ask the following question. Given a rearrangement invariant space
E on [0, 1], when is the natural embedding E ⊂ L
([0, 1]) strictly singular. (We
refer the reader to  for the deﬁnition and properties of rearrangement invariant
spaces.) We deﬁne a linear map between two normed spaces to be strictly singular
if there does not exist an inﬁnite dimensional subspace of the domain upon which
the operator is an isomorphism.
This question is a natural extension of similar work by del Amo, Hernández,
Sánchez and Semenov , when they considered the problem of which embed-
dings between rearrangement invariant spaces are not disjointly strictly singular. A
positive linear operator between two Banach lattices is disjointly strictly singular if
there exists an inﬁnite sequence of non-zero disjoint elements in the domain such
that the operator is an isomorphism on the span of this sequence. This work 
contains a number of very sharp results, giving some very clear criteria.
However the question concerning when such maps are strictly singular seems to
be more difﬁcult. For this reason, we will restrict ourselves to considering the case
when the range is L
([0, 1]). Even then, we do not have complete answers, and in
this paper, we leave as many questions unanswered as we answer.
Research supported in part by grants from the N.S.F. and the Research Board of the University
Research supported in part by a grant from the N.S.F. and a grant from Russia.