The Local spectral theory and surjective spectrum of linear relations

M. Mnif; A.-A. Ouled-Hmed

doi:10.37863/umzh.v73i2.81

M. Mnif Univ. Sfax, Tunisia https://orcid.org/0000-0002-0984-7481
A.-A. Ouled-Hmed Univ. Sfax, Tunisia

DOI: https://doi.org/10.37863/umzh.v73i2.81

Keywords: Linear relations, local spectrum, surjective spectrum, correlation analytic core, local and glocal spectral subspaces

Abstract

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 $K\prime (T)$ and quasinilpotent part $H_0(T)$ of a linear relation $T$ in a Banach space $X$. As an application, we give a characterization of the surjective spectrum $\sigma_{su}(T)$ in terms of the local spectrum and show that if $X = H_0(\lambda I - T) + K\prime (\lambda I - T)$, then $\sigma_{su}(T)$ does not cluster at $\lambda$.

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