Local duality for Banach spaces

A local dual of a Banach space X is a subspace of the dual X* which can replace the whole dual space when dealing with finite dimensional subspaces. This notion arose as a development of the principle of local reflexivity, and it is very useful when a description of X* is not available. We give an exposition of the theory of local duality for Banach spaces, including the main properties, examples and applications, and comparing the notion of local dual with some other weaker properties of the subspaces of the dual of a Banach space. © 2014 Elsevier Ltd.Research partially supported by DGI (Spain), Grant MTM2010-20190

Examples of tauberian operators acting on C[0, 1]

ABSTRACT:We show that some counterexamples in the theory of tauberian operators can be realized as operators acting on C[0, 1]. Precisely, we show that the set t(C[0, 1]) of tauberian operators acting on C[0, 1] is not open, and that T?t(C[0, 1]) does not imply T** tauberian

Classes of operators preserved by extensions or liftings

A standard way to obtain extensions (resp. liftings) of operators is by making the so-called operations of push-out (resp. pull-back). In this paper we study the preservation of some classes of operators associated with an operator ideal Aunder push-out extensions or pull-back liftings. We show several examples of classical operator ideals whose associated classes are preserved, we prove that the preservation of those classes under push-out extension or pull-back lifting implies that the space ideal of Asatisfies the 3-space property, and we derive some results for Athat are useful in the study of commutative diagrams of operators.that of the three authors has been supported in part by MINECO (Spain), Project MTM2016-76958

Cotauberian Operators on L1(0, 1) Obtained by Lifting

ABSTRACT:We show that the set Td(L1(0, 1)) of cotauberian operators acting on L1(0, 1) is not open, and T ? Td(L1(0, 1)) does not imply T** cotauberian. As a consequence, we derive that the set T(L8(0, 1)) of tauberian operators acting on L8(0, 1) is not open, and that T ? T(L8(0,1)) does not imply T** tauberian

Tauberian operators on L1(μ)L_1(μ) spaces

We characterize tauberian operators $T:L_1(μ) → Y$ in terms of the images of disjoint sequences and in terms of the image of the dyadic tree in $L_1[0,1]$. As applications, we show that the class of tauberian operators is stable under small norm perturbations and that its perturbation class coincides with the class of all weakly precompact operators. Moreover, we prove that the second conjugate of a tauberian operator $T:L_1(μ) → Y$ is also tauberian, and the induced operator $T̃: L_1(μ)**/L_1(μ) → Y**/Y$ is an isomorphism into. Also, we show that $L_1(μ)$ embeds isomorphically into the quotient of $L_1(μ)$ by any of its reflexive subspaces

On basic sequences in dual Banach spaces

AbstractA short proof of the following result is given: for every semi-normalized sequence (xn∗) in a dual Banach space X∗ with 0∈{xn∗}¯w∗, there exists a bounded sequence (xn) in X and a basic subsequence (xkn∗) such that 〈xki∗,xj〉=δij

Supertauberian operators and perturbations

Upper semi-Fredholm operators and tauberian operators in Banach spaces admit the following perturbative characterizations [6], [2]: An operator T: X --> Y is upper semi-Fredholm (tauberian) if and only if for every compact operator K: X --> Y the kernel N(T+K) is finite dimensional (reflexive). In [7] Tacon introduces an intermediate class between upper semi-Fredholm operators and tauberian operators, the supertauberian operators, and he studies this class using non-standard analysis. Here we study supertauberian operators using ultrapower of Banach spaces and, among other results, we obtain a perturbative characterization. As a consequence we characterize Banach spaces in which all superreflexive subspaces are finite dimensional, and Banach spaces in which all reflexive subspaces are superreflexive. Similar results are obtained for the dual class of cosupertauberian operators, including a perturbative characterization of this class, and characterizations of Banach spaces in which all quotients are finite dimensional, and Banach spaces in which all reflexive quotients are superreflexive

Quotients of L1 by reflexive subspaces

Here we present and example and some results suggesting that there is no infinite-dimensional reflexive subspace Z of L1 = L1[0,1] such that the quotient L1/Z is isomorphic to a subspace of L1