Uniformly factorizing analytic collections of weakly compact operators | Department of Mathematics

Uniformly factorizing analytic collections of weakly compact operators

Event Information
Event Location: 
GAB 461, 4-5 PM; Refreshments: GAB 472, 3:30 PM
Event Date: 
Monday, October 8, 2012 - 4:00pm

The famous Davis-Figiel-Johnson-Pelczynski factorization theorem from 1974 asserts that for any weakly compact operator $T: X \to Y$ there is a reflexive Banach space $Z$ such that $T=SR$ where $R:X \to Z$ and $S:Z \to Y$ bounded linear operators (in brief we say $T$ factors through $Z$). In this talk, we will show how by combining recent results that lie at the intersection of descriptive set theory and Banach space theory one can obtain an much stronger uniform version of this result.

Namely: If $A$ and $B$ are analytic collections of Banach separable Banach spaces, $B$ contains only spaces with shrinking basis and for each $X \in A$ and $Y \in B$ we have an analytic collection $A_{X,Y}$ of weakly compact operators from $X$ to $Y$, then there is a single reflexive space $Z$ such that every operator $T \in A_{X,Y}$ with $X \in A$ and $Y \in B$ factors through. We will give a basic outline of the proof and discuss consequences of this theorem.