LEFT-RIGHT BROWDER LINEAR RELATIONS AND RIESZ PERTURBATIONS

doi:10.1016/S0252-9602(17)30083-8

Abstract

Abstract:

A closed linear relation T in a Banach space X is called left (resp. right) Fredholm if it is upper (resp. lower) semiFredholm and its range (resp. null space) is topologically complemented in X. We say that T is left (resp. right) Browder if it is left (resp. right) Fredholm and has a finite ascent (resp. descent). In this paper, we analyze the stability of the left (resp. right) Fredholm and the left (resp. right) Browder linear relations under commuting Riesz operator perturbations. Recent results of Zivkovic et al. to the case of bounded operators are covered.

Key words: left and right Fredholm linear relations, left and right Browder linear relations, Riesz perturbations

Teresa ÁLVAREZ. LEFT-RIGHT BROWDER LINEAR RELATIONS AND RIESZ PERTURBATIONS[J].Acta mathematica scientia,Series B, 2017, 37(5): 1437-1452.

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References

[1] Álvarez T. Perturbation theorems for upper and lower semi-Fredholm linear relations. Publ Math Debrecen, 2004, 65:179-191
[2] Álvarez T. On regular linear relations. Acta Math Sin (English Ser), 2012, 28:183-194
[3] Álvarez T, Ammar A, Jeribi A. A characterization of some subs ets of S-essential spectra of a multivalued linear operator. Colloq Math, 2014, 135:171-186
[4] Álvarez T, Benharrat M. Relationship between the Kato spectrum and the Goldberg spectrum of a linear relation. Mediterr J Math, 2016, 13:365-378
[5] Álvarez T, Fakhfakh F, Mnif M. Left-right Fredholm and left-right Browder linear relations. Filomat, 2017, 31(2):255-271
[6] Cross R W. Multivalued Linear Operators. New York:Marcel Dekker, 1998
[7] Cross R W, Wilcox D. Multivalued linear projections. Quaestiones Math, 2002, 25:503-512
[8] Favini A, Yagi A. Multivalued operators and degenerate evolution equations. Ann Math Pura Appl, 1993, 163:353-384
[9] Goldmann M A, Krackovskii S N. Behaviour of the space of zero elements with finite-dimensional salient on the Riesz kernel under perturbations of the operators. Dokl Akad Nauk SSSR, 1975, 221:532-534(in Russian); English translation:Soviet Math Dokl, 1975, 16:370-373
[10] Jaftha J J. The conjugate of a product of linear relations. Comment Math Univ Carolin, 2006, 47:265-273
[11] Kaashoek M A. Ascent, descent, nullity and defect, a note on a paper by A. E. Taylor. Math Ann, 1967, 172:105-115
[12] Kaczynski T. Multivalued maps as a tool in modeling and rigorous numerics. J Fixed Point Theory Appl, 2008, 4:151-176
[13] von Neumann J. Functional Operators Ⅱ. The Geometry of Orthogonal Spaces. Ann Math Stud. Princeton:Princeton University Press, 1950
[14] Sandovici A, de Snoo H S V, Winkler H. Ascent, descent, nullity, defect, and related notions for linear relations in linear spaces. Lin Alg Appl, 2007, 423:456-497
[15] Sandovici A, de Snoo H S V. An index formula for the product of linear relations. Lin Alg Appl, 2009, 431:2160-2171
[16] Shvydkoy R, Latushkin Y. Operator algebras and the Fredholm spectrum of advective equations of linear hydrodynamics. Funct Anal, 2009, 257:3309-3328
[17] Taylor A E. Theorems on ascent, descent, nullity and defect of linear operators. Math Ann, 1966, 163:18-49
[18] Zivkovic-Zlatanovic S C, Djordjevic D S, Harte R E. On left and right Browder operators. J Korean Math Soc, 2011, 48(5):1053-1063
[19] Zivkovic-Zlatanovic S C, Djordjevic D S, Harte R E. Left-right Browder and left-right Fredholm operators. Integ Equ Oper Theory, 2011, 69:347-363
[20] Zivkovic-Zlatanovic S C, Cvetkovic M D, Djordjevic D. On closed upper and lower semi-Browder operators. Mediterr J Math, 2015, 12:1033-1045
[21] Zivkovic-Zlatanovic S C, Harte R E. Polynomially Riesz perturbations Ⅱ. Filomat, 2015, 29:2125-2136