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. Martínez-Abejón: 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.
