We initiate 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′(T) and the quasinilpotent part H0(T) of a linear relation T in a Banach space X. As an application, we present a characterization of the surjective spectrum 𝜎su(T) in terms of the local spectrum and show that if X = H0(⋋I − T) + K′(⋋I − T), then 𝜎su(T) does not cluster at ⋋.
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References
P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer AP, Dordrecht (2004).
P. Aiena, “Fredholm theory and localized SVEP,” Funct. Anal. Approx. Comput., 7, No. 2, 9–58 (2015).
E. Chafai, Ascent, Descent and Some Perturbation Results for Linear Relation: Doctor. Thesis, Univ. Sfax (2013).
E. Chafai and M. Mnif, “Perturbation of normally solvable linear relations in paracomplete space,” Linear Algebra Appl., 439, 1875–1885 (2013).
R. W. Cross, “Multivalued linear operators,” Monographs and Textbooks in Pure and Applied Mathematics, 213, Marcel Dekker, New York (1998).
N. Dunford, “Spectral theory. II. Resolution of the identity,” Pacific J. Math., 2, 559–614 (1952).
N. Dunford, “Spectral operators,” Pacific J. Math., 4, 321–354 (1954).
K. B. Laursen and M. M. Neuman, “An introduction to local spectral theory,” London Mathematical Society Monographs. New Series, 20, Clarendon Press, Oxford University Press, New York (2000).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 2, pp. 222–237, February, 2021. Ukrainian DOI: 10.37863/umzh.v73i2.81.
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Mnif, M., Ouled-Hmed, AA. Local Spectral Theory and Surjective Spectrum of Linear Relations. Ukr Math J 73, 255–275 (2021). https://doi.org/10.1007/s11253-021-01920-3
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DOI: https://doi.org/10.1007/s11253-021-01920-3