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

Ryan Causey
Published: 15 July 2022

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

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