Localization in the spectral theory of operators on Banach spaces

The Kato-type spectrum and local spectral theory

“…An "all or nothing" relation between SVEP and components of ̺ K (T ) was observed in [11,Theorem 13]. The following proposition provides a simple proof of this and a slight extension of Theorem 19, [11], as well.…”

“…The following proposition provides a simple proof of this and a slight extension of Theorem 19, [11], as well. The second statement is, in fact, a special case of Theorem 2.5, our main result of this section.…”

“…The present article centers around certain localized versions of some basic concepts of local spectral theory, with emphasis on decomposability in the sense of Foiaş and on Bishop's property (β); see [3], [5], [9], [11], [12], and [14]. An operator T ∈ L (X) is said to be decomposable on an open subset U of provided that, for every finite open cover {V 1 , .…”

“…The injectivity issue is addressed by the classical single-valued extension property (SVEP), [1] and [11]. An operator T ∈ L (X) is said to have SVEP at a point λ ∈ provided that, for every open disc V centered at λ, the mapping T V is injective on H(V, X).…”

Essential Spectral Inclusions for Operators on Banach Spaces

“…If T * has (SVEP) at 0, then T * 1 does as well, and since T * 1 is semi-regular, it follows by [12,Theorem 13] …”