On the local spectral theory for positive and non-positive operators
We discuss the local spectral behaviour of a continuous linear operator \(T\) defined on a Banach space by means of a system of \(T\)-invariant not necessarily closed subspaces. A similar description was given by the author for a positive linear operator \(T\) on a Banach lattice \(E\) with the aid of a chain of \(T\)-invariant order ideals. Here we give a slightly different definition and get a system of \(T\)-invariant subspaces which is maximal in some sense. It turns out that many of the results which are valid for positive operators on a Banach lattice hold also in similar forms in this general situation. But, of course, for positive operators there are a lot of additional informations. We apply this method to deduce some well-known results of the local spectral theory. Our method is elementary and uses only some simple facts about the theory of holomorphic vector valued functions, and we do not assume the operators to have the single valued extension property.
local spectral behaviour of a continuous linear operator
scientific article; zbMATH DE number 57314