Frechet-valued measures and nuclearity

• Autores: M.A. Sofi
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 33, Nº 4, 2007, págs. 1079-1090
• Idioma: inglés
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• Resumen

  • The recognition of sequences localised inside the range of a vector measure is an important theme in vector measure theory. In a previous work, the author had characterized Banach spaces X in which absolutely p-summable sequences in X (p>1) are contained inside the range of an X-valued measure of bounded variation precisely as those having (q)-Orlicz property (q being conjugate to p)- a property that characterizes X as finite dimensional as long as p > 2. This motivates the natural question of investigating this property in the setting of Frechet spaces where it is shown to translate into nuclearity -in conformity with the philosophy that nuclear Frechet spaces are better equipped to be called infinite-dimensional variants of finite dimensional spaces than are the more familiar Hilbert spaces. This result provides a strengthening of an earlier result of Bonet and Madrigal, characterising nuclearity of a Frechet space X with absolutely p- summable sequences in X being replaced by null sequences in X. The paper concludes with another useful and more general version of the said result in terms of (p,q)- summing multipliers.