The Local spectral theory and surjective spectrum of linear relations

UDC 517.98 This paper initiates a study of local spectral theory for linear relations. At the beginning, we define the local spectrum and study its properties. Then we obtain results related to the correlation analytic core and quasinilpotent part of a linear relation in a Banach space . As an application, we give a characterization of the surjective spectrum in terms of the local spectrum and show that if , then does not cluster at .

Further characterizations of property (VΠ) and some applications

We carry out characterizations with techniques provided by the local spectral theory of bounded linear operators T ∈ L(X), X infinite dimensional complex Banach space, which verify property (VΠ) introduced by Sanabria et al. (Open Math. 16(1) (2018), 289-297). We also carry out the study for polaroid operators and Drazin invertible operators that verify the property mentioned above.

Local Spectral Theory for R and S Satisfying RnSRn = Rj

In this paper, we analyze local spectral properties of operators R,S and RS which satisfy the operator equations RnSRn=Rj and SnRSn=Sj for same integers j≥n≥0. We also continue to study the relationship between the local spectral properties of an operator R and the local spectral properties of S. Thus, we investigate the transmission of some local spectral properties from R to S and we illustrate our results with an example. The theory is exemplified in some cases.

Decomposability of Krein space operators

In this paper, we review some properties in the local spectral theory and various subclasses of decomposable operators. We prove that every Krein space selfadjoint operator having property (?) is decomposable, and clarify the relation between decomposability and property (?) for J-selfadjoint operators. We prove the equivalence of these properties for J-selfadjoint operators T and T* by using their local spectra and local spectral subspaces.