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) se-quences 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 character-ize various classes of operators between Banach spaces, such as the weakly compact, Tauberian, and super-Tauberian oper-ators, 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 un-der 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 ar-rays result characterizing reflexivity, due to R.C. James. It i