On wide-(<i>s</i>) sequences and their applications to certain classes of operators

作     者：Rosenthal, H 

作者机构：Univ Texas Austin TX 78712 USA 

出 版 物：《PACIFIC JOURNAL OF MATHEMATICS》 (Pac. J. Math.)

年 卷 期：1999年第189卷第2期

页      面：311-338页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：Directorate for Mathematical and Physical Sciences  MPS  (9500874) 

主　　题：.weakly compact exivity basic sequence triangular arrays classes of operators WIDE SEQUENCES non-strongly Tauberian 

摘      要：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 epsilon 0, it maps some (1 + epsilon)-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 epsilon 0, it maps some (1 + epsilon)-wide-(s) sequence into a norm-convergent sequence (resp. a sequence whose image has diameter less than epsilon). 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. Gonzalez and A. Martinez-Abejon: An operator is non-super-Tauberian iff there are for every epsilon 0, finite (1 + epsilon)-wide-(s) sequences of arbitrary length whose images have norm at most epsilon.