DOI

We study properties of bounded sets in Banach spaces, connected with the concept of equimeasurability introduced by A. Grothendieck. We introduce corresponding ideals of operators and find characterizations of them in terms of continuity of operators in certain topologies. The following result (Corollary 9) follows from the basic theorems: Let T be a continuous linear operator from a Banach space X to a Banach space Y. The following assertions are equivalent: 1) T is an operator of type RN; 2) for any Banach space Z, for any number p, p > 0, and any p-absolutely summing operator U:Z → X the operator TU is approximately p-Radonifying; 3) for any Banach space Z and any absolutely summing operator U:Z → X the operator TU is approximately 1-Radonifying. We note that the implication I)⇒2), is apparently new even if the operator T is weakly compact.

Original languageEnglish
Pages (from-to)2156-2159
Number of pages4
JournalJournal of Soviet Mathematics
Volume34
Issue number6
DOIs
StatePublished - Sep 1986

    Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Applied Mathematics

ID: 73500514