Szlenk indices of operators between Banach spaces

Abstract: The notion of fragmentability often arises in situations where a set $K$ is equipped with both a topology $\tau$ and a (usually incompatible) metric $d$. One says that $(K, \tau)$ is fragmentable by $d$ if for every nonempty subset $L$ of $K$ and every $\epsilon >0$ there exists a nonempty $\tau$-open subset $U$ of $K$ such that $U \cap L$ is nonempty and the diameter of $U \cap L$ (with respect to $d$) does not exceed $\epsilon$.

The setting of the talk shall be an instance where the notion of fragmentability arises in the theory of Banach spaces and their operators. In particular, we shall consider the case where $K$ is a bounded subset of the dual of a Banach space, $\tau$ is the relative $w^\ast$-topology on $K$ and $d$ is the norm on the dual space; if an operator $T$ acting between Banach spaces has the property that its adjoint operator maps bounded sets into sets that are fragmentable in the above sense, then (via a process to be described in detail during the talk) we assign to $T$ a well-defined ordinal called the Szlenk index of $T$, denoted $Sz(T)$.

After giving some basic examples, and being motivated by the problem of classifying the closed ideals of operators on certain classical Banach spaces, the speaker shall discuss various aspects of the operator theory related to the Szlenk index. We mention here a sample result: for a given Banach space $X$ and ordinal $\alpha$, the set of all operators from $X$ to $X$ with $Sz(T) \leq \alpha$ is a closed, two-sided ideal in $B(X)$, the Banach algebra of all operators acting from $X$ to $X$.