{
  "id": "math/9610209",
  "version": "v1",
  "published": "1996-10-02T00:00:00.000Z",
  "updated": "1996-10-02T00:00:00.000Z",
  "title": "On wide-$(s)$ sequences and their applications to certain classes of operators",
  "authors": [
    "Haskell P. Rosenthal"
  ],
  "categories": [
    "math.FA"
  ],
  "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 $\\varepsilon$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1996-10-02T00:00:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B15"
    ],
    "keywords": [
      "super-tauberian operators",
      "banach space",
      "applications",
      "triangular arrays result characterizing reflexivity",
      "quantitative triangular arrays result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1996math.....10209R"
    }
  }
}