An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree $\mathcal T$ of a certain type on a space X is presumed to have a branch with some property. It is shown that then X can be embedded into a space with an FDD $(E_i)$ so that all normalized sequences in X which are almost a skipped blocking of $(E_i)$ have that property. As an application of our work we prove that if X is a separable reflexive Banach space and for some $1 0$, there exists a finite codimensional subspace of X which $C^2+\epsilon$ embeds into the $\ell_p$ sum of finite dimensional spaces.

Trees and Branches in Banach Spaces

This paper deals with the following types of problems: Assume a Banach space X has some property (P). Can it be embedded into some Banach space Z with a finite dimensional decomposition having property (P), or more generally, having a property related to (P)? Secondly, given a class of Banach spaces, does there exist a Banach space in this class, or in a closely related one, which is universal for this class?

/pdf/embedding-into-banach-spaces-with-finite-dimensional-5077dmpk5x.pdf

Embedding into Banach spaces with finite dimensional decompositions

We show that in Orlicz sequence spaces of Banach-Saks type p, weakly null sequences admit of p-Hilbertian subsequences. W. B. Johnson introduced the following notion in J]. Let 1 < p 1. A Banach space has Banach-Saks type p (property (BS p), for short), if every weakly null sequence has a subsequence (x k) so that for some C < 1 n X k=1 x k Cn 1=p for all n 2 IN: (Here, n 1=1 := 1 for all n 2 IN.) The following stronger property was introduced in KO]: Let 1 < p 1. A Banach space has property (S p), if every weakly null sequence has a subsequence (x k) so that for some C < 1 n X k=1 a k x k C for all n 2 IN and for all scalars (a k) with (n X k=1 ja k j p) 1=p 1: (Here, (P ja k j 1) 1=1 := maxja k j.) A sequence (x k), which is dominated by the unit vector basis of`p , i.e., which satisses the estimate in Deenition 2, is called p-Hilbertian. It follows from Elton's c 0-theorem (E], see also O], Corollary 4.4) that property (BS 1) implies property (S 1). Moreover, both properties are equivalent to the hereditary Dunford-Pettis property. For 1 < p < 1, however, both properties

Orlicz sequence spaces of Banach-Saks type

We prove that the dual or any quotient of a separable reflexive Banach space with the unconditional tree property (UTP) has the UTP. This is used to prove that a separable reflexive Banach space with the UTP embeds into a reflexive Banach space with an unconditional basis. This solves several longstanding open problems. In particular, it yields that a quotient of a reflexive Banach space with an unconditional finite-dimensional decomposition (UFDD) embeds into a reflexive Banach space with an unconditional basis

/pdf/a-characterization-of-subspaces-and-quotients-of-reflexive-1dlwpvj0x8.pdf

A characterization of subspaces and quotients of reflexive banach spaces with unconditional bases

Given a subspace X of lp, 1 < p < oo, the compact operators on X are an M-ideal in the bounded linear operators on X if and only if X has the compact approximation property. 0. Introduction. Recently Harmand and Lima [7] proved that if A" is a Banach space for which K(X), the space of compact operators on X, is an M-ideal in L(X), the space of bounded linear operators on X, then there is a net {7a} in K(X) so that: (i) Ta -» 7 strongly, '(ii)||7a||^lforalla, (iii) T* -* I strongly, (iv) lim.H/r„|| = l. The main result of this paper is a strong converse to the Harmand-Lima theorem for subspaces of lp, 1 < p < oo. In Theorem 6 we show that if X is a subspace of (LXn)p (dim A'„<oo;l</'<oo) which has the compact approximation property, then K(X) is an M-ideal in L( X). Part of the proof consists in showing that such an X satisfies conditions (i)-(iv) in the Harmand-Lima theorem. This result (which is simple given the state-of-the-art in Banach space theory) is proved for general reflexive spaces in §2. §3 is devoted to proving the converse of the Harmand-Lima theorem for subspaces of (LXn)p. Here we use blocking methods which have been previously used in the study of isomorphic, rather than isometric, properties of lp and a few other spaces. 1. Notation and preliminaries. If X and Y are Banach spaces, L(X, Y) (resp. K(X, Y)) will denote the space of all bounded linear operators (resp. compact linear operators) from X to Y. If X = Y then we simply write L(X) (resp. K(X)). Ball(X) will denote the closed unit ball of X. Received by the editors April 2, 1984. 1980 Mathematics Subject Classification. Primary 46A32, 47B05; Secondary 47B05, 41A50.

https://www.ams.org/proc/1985-093-03/S0002-9939-1985-0774004-5/S0002-9939-1985-0774004-5.pdf

A characterization of subspaces of _{} for which () is an -ideal in ()

On quotients of $L_p$ which are quotients of $\ell _p$

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On quotients of $L_p$ which are quotients of $\ell _p$