Lineability of the set of bounded linear non-absolutely summing operators

Abstract

In this note we solve, except for extremely pathological cases, a question posed by Puglisi and Seoane-Sepúlveda on the lineability of the set of bounded linear non-absolutely summing operators. We also show how the idea of the proof can be adapted to several related situations.

Operator ranges and spaceability

“…One of our results will be an improvement (Theorem 4.1) on recent results of Botelho et al [5,6], which considered a problem raised in [22] about lineability of the complement B(X, Y ) \ Π p (X, Y ) of the p-summing operators between Banach spaces X and Y (1 p < ∞). We can establish spaceability of the intersection of these complements, under more general conditions on X and Y .…”

“…The question posed in [22,Problem 2.4] was whether super-reflexivity of X (and infinite dimension of Y ) is sufficient to ensure lineability of B(X, Y ) \ Π p (X, Y ) for each p. Botelho et al [5,6] obtained positive answers using conditions relating to existence of subspaces of X or of Y with unconditional basis (and required that the subspace be complemented in the case of X ). In fact they obtained a subspace in (K(X, Y ) \ Π p (X, Y )) ∪ {0} vector space isomorphic to 1 (hence of uncountable dimension) in [6].…”

The history of a general criterium on spaceability

“…The following year, and partially answering a question posed in [14], Botelho, Diniz and Pellegrino proved in [15] that if E is a superreflexive Banach space containing a complemented infinite-dimensional subspace with unconditional basis, or F is a Banach space having an infinite unconditional basic sequence, then the set K.E; F / n … p .E; F / is lineable for every p 1, where K denotes the ideal of all compact operators.…”

Sierpiński-Zygmund functions and other problems on lineability

“…Also, in [5], the authors constructed, given any set E ⊂ T of Lebesgue measure zero, an infinite-dimensional, infinitely generated dense subalgebra of C(T) every non-zero element of which has a Fourier series expansion divergent in E. More authors have been working on these types of problems related to lineability and algebrability in the framework of vector measures, operator theory, polynomials, and holomorphy. (We refer to, e.g., [1,7,8,10,11,21,22] for some recent advances in this theory. )…”