On the bounded approximation property in Banach spaces

Bulletin of the Polish Academy of Sciences Mathematics

Let E be a Banach lattice and let X be its closed subspace such that: X is complemented in E, or the norm of E is order continuous. Then X is reflexive iff X* contains no isomorphic copy of \ell_1 iff for every n ≥ 1, the nth dual X^(n) of X contains no isomorphic copy of \ell_1 iff X has no quotient isomorphic to c_0 and X does not have a subspace isomorphic to \ell_1 (Theorem 2). This is an extension of the results obtained earlier by Lozanovski˘ i, Tzafriri, Meyer-Nieberg, and Diaz and Fern´andez. The theorem is applied to show that many Banach spaces possess separable quotients isomorphic to one of the following spaces: c_0, \ell_1, or a reflexive space with a Schauder basis.

2002

1. If X is a Banach space of type p and of cotype q, then every its n-dimensional subspace is Cn 1/p−1/q-complemented in X (cf. [1]). Szankowski [2] has showed that if T(X) = sup{p: X of type p} ̸ = 2 or C(X) = inf{q: X of cotype q} ̸ = 2, then X has a subspace without the approximation property. Thus, if each subspace of X possesses the approximation property, then necessarily T(X) = C(X) = 2 and, therefore, all of finite dimensional subspaces in X are ”well ” complemented. Moreover, the space X need not be Hilbertian (or isomorphic to a Hilbert space). The examples of such spaces were constructed by Johnson [3]. In connection with the examples of Szankowski and Johnson, the natural questions arises: 1)if T(X) = C(X) = 2, then is it true that every subspace in X has the approximation property? 2) how ”well complemented ” may be the finite dimensional subspaces of a space without the approximation property: in particular, does there exist a space X without the approximation property...

The notion of a basis in a Banach space was introduced by J. Schauder in 1927. If a Banach space has a basis then it is also separable. The problem whether every separable Banach space has a Schauder basis appeared for the first time in 1931 in Banach’s book "Theory of Linear Operations". It was clear to Banach, Mazur and Schauder that this question was related to the "approximation problem". This is mentioned in Banach’s book as a remark to the chapter on compact operators. If a Banach space X has a Schauder basis it also has the approximation property, since the natural projections of X onto its finite dimensional subspaces form a bounded sequence of finite rank operators converging pointwise on X to the identity operator. The approximation problem is equivalent to whether a Banach space has the "approximation property". A Banach space X has the approximation property if the identity operator on X is the limit for the topology of uniform convergence on compact subsets of X of a sequence of finite rank operators. The approximation property in its various forms was thoroughly analyzed by A. Grothendieck in his thesis "Produits tensoriels topologiques et espaces nucléaires" published in 1955. But it was not until 1972, that P. Enflo solved both questions in the negative. He found a subspace of co, which he showed does not have the approximation property and consequently does not have a basis. Almost immediately, T. Figiel and A. M. Davie greatly simplified his proof. The later using a probabilistic lemma constructed a separable closed subspace of l1 without the approximation property. Moreover, they also showed that both co and every lp space with 1  p < 1 , (p 6= 2) has subspaces without the approximation property. In this work we present some of the equivalent properties to the approximation property due to A. Grothendieck, and we make a detailed exposition of the proof by A. M. Davie.

1997

In this survey we shaw that the separable quotient problem for Banach spac~in equivalent to several other problema from Banach space theory. Wc give afro several partial solutiona to the prob1cm.

Functional Analysis and Its Applications, 1983

1. If X is a Banach space of type p and of cotype q, then every its n-dimensional subspace is Cn 1/p−1/q -complemented in X (cf.

Mediterranean Journal of Mathematics, 2014

In this paper we characterize compact linear operators from Banach function spaces to Banach spaces by means of approximations with bounded homogeneous maps. To do so, we undertake a detailed study of such maps, proving a factorization theorem and paying special attention to the equivalent strong domination property involved. Some applications to compact maximal extensions of operators are also given. Banach function space and p-th power and compact operator and homogeneous operator. 46E30 and 47B38 and 46B42 and 46B28

2016

Central European Journal of Mathematics, 2011

Let X , Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R : X → Y such that its range, R(X ), is dense in Y .

Forum Mathematicum, 2019

This paper studies the bounded approximation property (BAP) in quasi-Banach spaces. In the first part of the paper, we show that the kernel of any surjective operator {\ell_{p}\to X} has the BAP when X has it and {0

2010

It is a translation of an old paper of mine. We describe the topology tau_p in the space Pi_p(Y,X), for which the closures of convex sets in tau_p and in *-weak topology of the space Pi_p(Y,X) are coincident. Thereafter, we investigate some properties of the space Pi_p, related to this new topology. 2010-remark: Occasionally, the topology is coincides with the lambda_p-topology from the paper &quot;Compact operators which factor through subspaces of l_p&quot;, Math. Nachr. 281(2008), 412-423 by Deba Prasad Sinha and Anil Kumar Karn.