SPECTRAL MAPPING THEOREM FOR ASCENT, ESSENTIAL ASCENT, DESCENT AND ESSENTIAL DESCENT SPECTRUM OF LINEAR RELATIONS

doi:10.1016/S0252-9602(14)60080-1

Abstract

Abstract:

In [7], Cross showed that the spectrum of a linear relation T on a normed space satisfies the spectral mapping theorem. In this paper, we extend the notion of essential ascent and descent for an operator acting on a vector space to linear relations acting on Banach spaces. We focus to define and study the descent, essential descent, ascent and essential ascent spectrum of a linear relation everywhere defined on a Banach space X. In particular, we show that the corresponding spectrum satisfy the polynomial version of the spectral mapping theorem.

Key words: ascent, essential ascent, descent, essential descent, linear relations

CLC Number:

47A06

Ezzeddine CHAFAI, Maher MNIF. SPECTRAL MAPPING THEOREM FOR ASCENT, ESSENTIAL ASCENT, DESCENT AND ESSENTIAL DESCENT SPECTRUM OF LINEAR RELATIONS[J].Acta mathematica scientia,Series B, 2014, 34(4): 1212-1224.

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References

[1] Alvarez T. On the Browder essential spectrum of a linear relation. Publ Math Debrecen, 2008, 75(1/2): 145–154

[2] Alvarez T. Small perturbation of normally solvable relations. Publ Math Debrecen, 2012, 80(1/2): 155–168

[3] Fredj O B H. Essential descent spectrum and commuting compact perturbations. Extracta Mathematicae, 2006, 21(3): 261–271

[4] Burgos M, Kaidi A, Mbekhta M, Oudghiri M. The descent spectrum and perturbations. J Operator Theory, 2006, 56(2): 259–271

[5] Coddington E A. Multivalued Operators and Boundary Value Problems. Lecture Notes in Math 183. Berlin: Springer-Verlag, 1971

[6] Cross R W. An index theorem for the product of linear relations. Linear Alg Appl, 1998, 277: 127–134

[7] Cross R W. Multivalued Linear Operators. New York: Marcel Dekker, 1998

[8] Fakhfakh F, Mnif M. Perturbtion theory of lower semi-Browder multivalued linear operators. Publ Math Debrecen, 2011, 78(3/4): 595–606

[9] Grabiner S. Uniform ascent and descent of bounded operators. J Math Soc Japan, 1982, 34: 317–337

[10] Grabiner S, Zemanek J. Ascent, descent and ergodic properties of linear operators. J Operator Theory, 2002, 48: 69–81

[11] Muler V. Spectral theory of linear operators and spectral systems in Banach algebras. Operator Theory: Advances and Applications Vol 139. Berlin: Springer, 2007

[12] Von Neumann J. Functional Operators II, The Geometry of Orthogonal Spaces. Annals of Math Studies. Princeton N J: Princeton University Press, 1950

[13] Sandovici A, de Snoo H, Winkler H. Ascent, descent, nullity, defect, and related notions for linear relations in linear spaces. Linear Alg Appl, 2007, 423: 456–497

[14] Sandovici A. Some basic properties of polynomials in a linear relation in linear spaces//Oper Theory Adv Appl 175. Basel: Birkhauser, 2007: 231–240

[15] Sandovici A, de Snoo H. An index formula for the product of linear relations. Linear Alg Appl, 2009, 431: 2160–2171

[16] Taylor A E. Theorems on ascent, descent, nullity and defect of linear operators. Math Ann, 1966, 163: 18–49