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 quantified versions, termed λ-wide-(s) sequences, are used to characterize various classes of operators between Banach spaces, such as the weakly compact, Tauberian, and super-Tauberian operators, as well as a new intermediate class introduced here, the strongly Tauberian operators. This is a nonlocalizable class which nevertheless forms an open semigroup and is closed under natural operations such as taking double adjoints. It is proved for example that an operator is non-weakly compact iff for every ε > 0, it maps some (1+ε)-wide-(s)-sequence to a wide-(s) sequence. This yields the quantitative triangular arrays result characterizing reflexivity, due to R.C. James. It is shown that an operator is non-Tauberian (resp. non-strongly Tauberian) iff for every ε > 0, it maps some (1 + ε)-wide-(s) sequence into a norm-convergent sequence (resp. a sequence whose image has diameter less than ε). This is applied to obtain a direct “finite” characterization of super-Tauberian operators, as well as the following characterization, which strengthens a recent result of M. González and A. Mart́ınezAbejón: An operator is non-super-Tauberian iff there are for every ε > 0, finite (1 + ε)-wide-(s) sequences of arbitrary length whose images have norm at most ε.