L. Tzafriri

On orlicz sequence spaces

It is proved that every Orlicz sequence space contains a subspace isomorphic to somelp. The question of uniqueness of symmetric bases in Orlicz sequence spaces is investigated.

On the complemented subspaces problem

A Banach space is isomorphic to a Hilbert space provided every closed subspace is complemented. A conditionally σ-complete Banach lattice is isomorphic to anLp-space (1≤p<∞) or toc0(Γ) if every closed sublattice is complemented.

Invertibility of ‘large’ submatrices with applications to the geometry of Banach spaces and harmonic analysis

AbstractThe main problem investigated in this paper is that of restricted invertibility of linear operators acting on finite dimensionallp-spaces. Our initial motivation to study such questions lies in their applications. The results obtained below enable us to complete earlier work on the structure of complemented subspaces ofLp-spaces which have extremal euclidean distance.LetA be a realn ×n matrix considered as a linear operator onlpn; l ≦p ≦ ∞. By restricted invertibility ofA, we mean the existence of a subset σ of {1, 2, …,n} such that |σ| ∼n andA acts as an isomorphism when restricted to the linear span of the unit vectorseii=1n There are various conditions under which this property holds. For instance, if the norm ‖A‖p ofA is bounded by a constant independent ofn and the diagonal ofA is the identity matrix, then there exists an index set σ, |σ| ∼n, for which (Rσ) has a bounded inverse σ stands for the restriction map). This is achieved by simply constructing the set σ so that ••Rσ(A-I)Rσ••p<21.The casep=2 is of particular interest. Although the problem is purely Hilbertian, the proofs involve besides the spacel2 also the spacel1. The methods are probabilistic and combinatorial. Crucial use is made of Grothendieck’s theorem.The paper also contains a nice application to the behavior of the trigonometric system on sets of positive measure, generalizing results on harmonic density. Given a subsetB of the circleT of positive Lebesgue measure, there exists a subset Λ of the integersZ of positive density dens Λ > 0 such that {fx137-1} whenever the support of the Fourier transform

On banach spaces with unconditional bases

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The uniform approximation property in Orlicz spaces

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On the type and cotype of Banach spaces

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Embeddingl p k in subspaces ofL p forp>2

off lies in Λ. The matrices involved here are Laurent matrices.The problem of restricted invertibility is meaningful beyond the class oflp-spaces, as is shown in a separate section. However, most of the paper uses specificlp-techniques and complete results are obtained only in the context oflp-spaces.

On multiplicity theory for Boolean algebras of projections

It is proved that the set ofps such thatlp is isomorphic to a subspace of a given Orlicz spacelFforms an interval. Some examples and properties of minimal Orlicz sequence spaces are presented. It is proved that an Orlicz function space (different froml2) is not isomorphic to a subspace of an Orlicz sequence space. Finally it is shown (under a certain restriction) that if two Orlicz function spaces are isomorphic, then they are identical (i.e. consist of the same functions).