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Disjointly strictly-singular inclusions between rearrangement invariant spaces

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García del Amo Jiménez, Alejandro José and Hernández, Francisco L. and Sánchez de los Reyes, Víctor Manuel and Semenov, Evgeny M. (2000) Disjointly strictly-singular inclusions between rearrangement invariant spaces. Journal London Mathematical Society, 62 (1). pp. 239-252. ISSN 1469-7750

Official URL: http://jlms.oxfordjournals.org/content/62/1/239.abstract


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Abstract

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]↪Lp[0, 1] for 1≤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.


Item Type:Article
Uncontrolled Keywords:Disjointly strictly-singular; Banach lattice; Inclusions; Rearrangement invariant (r.i.) function spaces; Characterizations of L 1 and L 1 among the r.i. function spaces
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:22843
Deposited On:18 Sep 2013 15:32
Last Modified:08 Nov 2013 17:33

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