The Local spectral theory and surjective spectrum of linear relations

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


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 $\lambda$.


P. Aiena, Fredholm and local spectral theory, with applications to multipliers, Kluwer Acad. Publ. (2004).

P. Aiena, Fredholm theory and localized SVEP, Funct. Anal., Approxim. and Comput., 7, № 2, 9 – 58 (2015).

E. Chafai, Ascent, descent and some perturbation results for linear relation: Doctor. Thesis, Univ. Sfax (2013).

E. Chafai, M. Mnif, Perturbation of normally solvable linear relations in paracomplete space, Linear Algebra and Appl., 439, 1875 – 1885 (2013), DOI:

R. W. Cross, Multivalued linear operators, Marcel Dekker, New York (1998).

N. Dunford, Spectral theory II. Resolution of the identity, Pacif. J. Math., 2, 559 – 614 (1952). DOI:

N. Dunford, Spectral operators, Pacific J. Math., 4, 321 – 354 (1954). DOI:

K. B. Laursen, M. M. Neuman, An introduction to local spectral theory, London Math. Soc.Monogr. 20, Clarendon Press, Oxford (2000).

How to Cite
Mnif, M., and A.-A. Ouled-Hmed. “The Local Spectral Theory and Surjective Spectrum of Linear Relations”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 2, Feb. 2021, pp. 222 -37, doi:10.37863/umzh.v73i2.81.
Research articles