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Gelfand numbers of operators with values in a Hilbert space

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In the paper we prove two inequalities involving Gelfand numbers of operators with values in a Hilbert space. The first inequality is a Rademacher version of the main result in [Pa-To-1] which relates the Gelfand numbers of an operator from a Banach spaceX intol n2 with a certain Rademacher average for the dual operator. The second inequality states that the Gelfand numbers of an operatoru froml N1 into a Hilbert space satisfy the inequality

$$k^{1/2} c_k (u) \leqq C\parallel u\parallel (\log (1 + N/k))^{1/2}$$

whereC is a universal constant. Several applications of these inequalities in the geometry of Banach spaces are given.

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Authors and Affiliations

  1. Kurt-Zier-Str. 8, DDR-6902, Jena, NLO

    Bernd Carl

  2. Equipe d'Analyse, Université Paris VI, Tour 46-4 ème Etage, 4 Plase Jussieu, F-75252, Paris Cedex 05, France

    Alain Pajor

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  1. Bernd Carl
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Carl, B., Pajor, A. Gelfand numbers of operators with values in a Hilbert space. Invent Math 94, 479–504 (1988). https://doi.org/10.1007/BF01394273

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