Francisco L. Hernández

Onl p-complemented copies in Orlicz spaces II

For anyp > 1, the existence is shown of Orlicz spacesLF andlF with indicesp containingsingular lp-complemented copies, extending a result of N. Kalton ([6]). Also the following is proved:Let 1 <α ≦β < ∞and H be an arbitrary closed subset of the interval [α, β].There exist Orlicz sequence spaces lF (resp. Orlicz function spaces LF)with indices α and β containing only singular lp-complemented copies and such that the set of values p > 1for which lp is complementably embedded into lF (resp. LF)is exactly the set H (resp. H ∪ {2}). An explicitly defined class of minimal Orlicz spaces is given.

Domination by positive disjointly strictly singular operators

We prove that each positive operator from a Banach lattice E to a Banach lattice F with a disjointly strictly singular majorant is itself disjointly strictly singular provided the norm on F is order continuous. We prove as well that if S : E --> E is dominated by a disjointly strictly singular operator, then S-2 is disjointly strictly singular.

DISJOINTLY STRICTLY-SINGULAR INCLUSIONS BETWEEN REARRANGEMENT INVARIANT SPACES

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.

Disjointly strictly singular operators and interpolation

Interpolation properties of the class of disjointly strictly singular operators on Banach lattices are studied. We also give some applications to compare the lattice structure of two rearrangement invariant function spaces. In particular, we obtain suitable analytic characterisations of when the inclusion map between two Orlicz function spaces is disjointly strictly singular.

Strictly singular operators on L_p spaces and interpolation

We study the class Vp of strictly singular non-compact operators on Lp spaces. This allows us to obtain interpolation results for strictly singular operators on Lp spaces. Given 1 ≤ p < q ≤ ∞, it is shown that if an operator T bounded on Lp and Lq is strictly singular on Lr for some p ≤ r ≤ q, then it is compact on Ls for every p < s < q.

Lattice-embedding scales ofLp spaces into orlicz spaces

We study the setPX of scalarsp such thatLp is lattice-isomorphically embedded into a given rearrangement invariant (r.i.) function spaceX[0, 1]. Given 0<α≤β<∞, we construct a family of Orlicz function spacesX=LF[0, 1], with Boyd indicesα andβ, whose associated setsPX are the closed intervals [γ, β], for everyγ withα≤γ≤β. In particular forα>2, this proves the existence of separable 2-convex r.i. function spaces on [0,1] containing isomorphically scales ofLp-spaces for different values ofp. We also show that, in general, the associated setPX is not closed. Similar questions in the setting of Banach spaces with uncountable symmetric basis are also considered. Thus, we construct a family of Orlicz spaces ℓF(I), with symmetric basis and indices fixed in advance, containing ℓp(Γ-subspaces for differentp’s and uncountable Λ⊂I. In contrast with the behavior in the countable case (Lindenstrauss and Tzafriri [L-T1]), we show that the set of scalarsp for which ℓp(Γ) is isomorphic to a subspace of a given Orlicz space ℓF(I) is not in general closed.

Lattice-embeddingLp into Orlicz spaces

Given 0<α≤p≤β<∞, we construct Orlicz function spacesLF[0, 1] with Boyd indicesα andβ such thatLp is lattice isomorphic to a sublattice ofLF[0, 1]. Forp>2 this shows the existence of (non-trivial) separable r.i. spaces on [0, 1] containing an isomorphic copy ofLp. The discrete case of Orlicz spaces ℓF (I) containing an isomorphic copy of ℓp(Γ) for uncountable sets Γ ⊂I is also considered.

Strictly Singular Embeddings

An operator A mapping a Banach space E into a Banach space F is called strictly singular (or Kato) if any restriction of A to an infinite-dimensional subspace of E is not an isomorphism. The paper deals with the problem of describing all couples of rearrangement-invariant spaces E↪F for which the embedding operator is strictly singular.

Disjointly Homogeneous Banach Lattices and Applications

This is a survey on disjointly homogeneous Banach lattices and their applications. Several structural properties of this class are analyzed. In addition we show how these spaces provide a natural framework for studying the compactness of powers of operators allowing for a unified treatment of well-known results.

Bernstein Widths and Super Strictly Singular Inclusions

The super strict singularity of inclusions between rearrangement invariant function spaces on [0, 1] is studied. Estimates of the Bernstein widths 𝛾n of the inclusions L∞ are given. It is showed that if the inclusion is strong and the order continuous part of exp E2 is not included in ⊂ then the inclusion ?? F is super strictly singular. Applications to the classes of Lorentz and Orlicz spaces are given.