The Type (B) Problem for Well-bounded Operators

 Tuesday, 21-March-2023 

Abstract

A scalar-type spectral operator T is an operator acting on a Banach space that can be represented as an integral over σ(T) with respect to a spectral measure. When the Banach space is reflexive, it had been shown by Dunford that T being scalar-type spectral is equivalent to T having a C(σ(T)) functional calculus. In 1994, Doust and deLaubenfels showed that this equivalence holds precisely on Banach spaces that do not have a subspace isomorphic to c0. A well-bounded operator T on a Banach space is an operator that has an AC[a, b] functional calculus, and it is said to be of type (B) if it can be written as an integral with respect to a spectral family of projections. On a reflexive Banach space, all well-bounded operators are of type (B). However, classifying the Banach spaces on which all well-bounded operators are of type (B) is still an open problem, and it has been conjectured by Doust and deLaubenfels that these are precisely the reflexive Banach spaces. In this talk, we will discuss the current standing of this problem and some recent progres