Tauberian operators on Banach spaces

Abstract

A Tauberian operator: E → F E \to F (Banach spaces) is one which satisfies T g ∈ F , g ∈ E Tg \in F,g \in E imply g ∈ E g \in E . The action of such operators and their pre-images on compact sets is studied in order to compare “Tauberian” with “weakly compact", an opposite property. Properties related to range closed are introduced which force operators with Tauberian-like properties to be Tauberian. Classes of spaces appear for which Tauberian is equivalent to semi-Fredholm. One example of this is the historical reason for the definition of these operators.