$AC(\sigma)$ --- the story so far

Abstract:

There is an extensive and classical theory concerning the structure of Hilbert or Banach space operators which admit a $C(\sigma(T))$ functional calculus. A similar but weaker theory was developed in the 1960s for operators for which one can form $f(T)$ for $f \in AC[a,b]$ (with $\Vert f(T) \Vert \le   K \Vert f \Vert_{BV}$). For the past 60 years the hunt has been on to see whether one can adapt this sensibly to overcome the restriction that this theory only applies to operators with real spectrum. In this talk I will discuss recent progress and the current state of play.