On wide-(s) sequences and their applications to certain classes of operators

A basic sequence in a Banach space is called wide-(s) if it is bounded and dominates the summing basis. (Wide-(s) sequences were originally introduced by I. Singer, who termed them P*-sequences.) These sequences and their qua ntified versions, termed lambda-wide-(s) sequences, are used to characteriz e various classes of operators between Banach spaces, such as the weakly co mpact, Tauberian, and super-Tauberian operators, as well as a new intermedi ate class introduced here, the strongly Tauberian operators. This is a nonl ocalizable class which nevertheless forms an open semigroup and is closed u nder natural operations such as taking double adjoints. It is proved for ex ample that an operator is non-weakly compact iff for every epsilon > 0, it maps some (1 + epsilon)-wide-(s)-sequence to a wide-(s) sequence. This yiel ds the quantitative triangular arrays result characterizing reflexivity, du e to R.C. James. It is shown that an operator is non-Tauberian (resp. non-s trongly Tauberian) iff for every epsilon > 0, it maps some (1 + epsilon)-wi de-(s) sequence into a norm-convergent sequence (resp. a sequence whose ima ge has diameter less than epsilon). This is applied to obtain a direct "fin ite" characterization of super-Tauberian operators, as well as the followin g characterization, which strengthens a recent result of M. Gonzalez and A. Martinez-Abejon: An operator is non-super-Tauberian iff there are for ever y epsilon > 0, finite (1 + epsilon)-wide-(s) sequences of arbitrary length whose images have norm at most epsilon.