Gelfand numbers of operators with values in a Hilbert space

SummaryIn 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 intol2n with a certain Rademacher average for the dual operator. The second inequality states that the Gelfand numbers of an operatoru froml1N 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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