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

Abstract

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 $\lambda$-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 $\varepsilon >0$, it maps some $(1+\varepsilon)$-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 $\varepsilon>0$, it maps some $(1+\varepsilon)$-wide-$(s)$ sequence into a norm-convergent sequence (resp. a sequence whose image has diameter less than $\varepsilon$). 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.~Gonz\'alez and A.~Mart{\'\i}nez-Abej\'on: An operator is non-super-Tauberian iff there are for every $\varepsilon>0$, finite $(1+\varepsilon)$-wide-$(s)$ sequences of arbitrary length whose images have norm at most $\varepsilon$.