Asymptotics of Operator Semigroups

Abraham Ng (University of Oxford)

Thursday 6th February, 2020 16:00-17:00 Maths 311B

Abstract

The well known Katznelson-Tzafriri theorem states that a power-bounded operator T on a Banach space X satisfies |T^n(I-T)| → 0 as n → ∞ if and only if the spectrum of T touches the complex unit circle nowhere except possibly at the point {1}. As it turns out, the rate at which |T^n(I-T)| goes to zero is largely determined by estimates on the resolvent of T on the unit circle minus {1} and not only is this interesting from a purely spectral and operator theoretic perspective, the applications of such quantified decay rates are myriad. In this talk, we will tell the twin stories of the so-called quantified Katznelson-Tzafriri theorems for discrete operator semigroups and the quantified Tauberian theorems for strongly continuous operator semigroups. Tracing through previously known results up to the present, we end with a new result that largely completes the discrete adventure.

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