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The Szlenk index of 𝐿_{𝑝}(𝑋) and 𝐴_{𝑝}

Ryan Causey

Abstract: Given a Banach space XX, a ww^*-compact subset of XX^*, and 1>p>1>p>\infty, we provide an optimal relationship between the Szlenk index of KK and the Szlenk index of an associated subset of Lp(X)L_p(X)^*. As an application, given a Banach space XX, we prove an optimal estimate of the Szlenk index of Lp(X)L_p(X) in terms of the Szlenk index of XX. This extends a result of Hájek and Schlumprecht to uncountable ordinals. More generally, given an operator A:XYA:X\to Y, we provide an estimate of the Szlenk index of the “pointwise AA” operator Ap:Lp(X)Lp(Y)A_p:L_p(X)\to L_p(Y) in terms of the Szlenk index of AA.
Keywords: math/mathml / Szlenk index / inline formula / Banach space / content type / formula content
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