On the stability of the localized single-valued extension property under commuting perturbations

Aiena, Pietro; Neumann, Michael

doi:10.1090/S0002-9939-2013-11635-7

On the stability of the localized single-valued extension property under commuting perturbations
HTML articles powered by AMS MathViewer

by Pietro Aiena and Michael M. Neumann PDF

Proc. Amer. Math. Soc. 141 (2013), 2039-2050 Request permission

Abstract:

This article concerns the permanence of the single-valued extension property at a point under suitable perturbations. While this property is, in general, not preserved under sums and products of commuting operators, we obtain positive results in the case of commuting perturbations that are quasi-nilpotent, algebraic, or Riesz operators.

References

Pietro Aiena, Fredholm and local spectral theory, with applications to multipliers, Kluwer Academic Publishers, Dordrecht, 2004. MR 2070395
Pietro Aiena, Quasi-Fredholm operators and localized SVEP, Acta Sci. Math. (Szeged) 73 (2007), no. 1-2, 251–263. MR 2339864
Pietro Aiena and Maria Teresa Biondi, Browder’s theorems through localized SVEP, Mediterr. J. Math. 2 (2005), no. 2, 137–151. MR 2184191, DOI 10.1007/s00009-005-0035-9
Pietro Aiena, Maria T. Biondi, and Carlos Carpintero, On Drazin invertibility, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2839–2848. MR 2399049, DOI 10.1090/S0002-9939-08-09138-7
Pietro Aiena, Carlos Carpintero, and Ennis Rosas, Some characterizations of operators satisfying $a$-Browder’s theorem, J. Math. Anal. Appl. 311 (2005), no. 2, 530–544. MR 2168416, DOI 10.1016/j.jmaa.2005.03.007
Pietro Aiena, Maria Luisa Colasante, and Manuel González, Operators which have a closed quasi-nilpotent part, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2701–2710. MR 1900878, DOI 10.1090/S0002-9939-02-06386-4
Pietro Aiena, T. Len Miller, and Michael M. Neumann, On a localised single-valued extension property, Math. Proc. R. Ir. Acad. 104A (2004), no. 1, 17–34. MR 2139507, DOI 10.3318/PRIA.2004.104.1.17
Pietro Aiena and Osmin Monsalve, Operators which do not have the single valued extension property, J. Math. Anal. Appl. 250 (2000), no. 2, 435–448. MR 1786074, DOI 10.1006/jmaa.2000.6966
Pietro Aiena and Osmin Monsalve, The single valued extension property and the generalized Kato decomposition property, Acta Sci. Math. (Szeged) 67 (2001), no. 3-4, 791–807. MR 1876467
M. Berkani and M. Sarih, On semi B-Fredholm operators, Glasg. Math. J. 43 (2001), no. 3, 457–465. MR 1878588, DOI 10.1017/S0017089501030075
A. Bourhim and V Miller G., The single-valued extension property is not preserved under sums and products of commuting operators, Glasg. Math. J. 49 (2007), no. 1, 99–104. MR 2337870, DOI 10.1017/S001708950700345X
Ion Colojoară and Ciprian Foiaş, Theory of generalized spectral operators, Mathematics and its Applications, Vol. 9, Gordon and Breach Science Publishers, New York-London-Paris, 1968. MR 0394282
Driss Drissi, The single-valued extension property and subharmonicity, Extracta Math. 22 (2007), no. 3, 235–244. MR 2405083
Nelson Dunford, Spectral theory. II. Resolutions of the identity, Pacific J. Math. 2 (1952), 559–614. MR 51435, DOI 10.2140/pjm.1952.2.559
Nelson Dunford, Spectral operators, Pacific J. Math. 4 (1954), 321–354. MR 63563, DOI 10.2140/pjm.1954.4.321
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III: Spectral operators, Pure and Applied Mathematics, Vol. VII, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1971. With the assistance of William G. Bade and Robert G. Bartle. MR 0412888
James K. Finch, The single valued extension property on a Banach space, Pacific J. Math. 58 (1975), no. 1, 61–69. MR 374985, DOI 10.2140/pjm.1975.58.61
Manuel González, Mostafa Mbekhta, and Mourad Oudghiri, On the isolated points of the surjective spectrum of a bounded operator, Proc. Amer. Math. Soc. 136 (2008), no. 10, 3521–3528. MR 2415036, DOI 10.1090/S0002-9939-08-09549-X
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020
Jean-Philippe Labrousse, Les opérateurs quasi Fredholm: une généralisation des opérateurs semi Fredholm, Rend. Circ. Mat. Palermo (2) 29 (1980), no. 2, 161–258 (French, with English summary). MR 636072, DOI 10.1007/BF02849344
Kjeld B. Laursen and Michael M. Neumann, An introduction to local spectral theory, London Mathematical Society Monographs. New Series, vol. 20, The Clarendon Press, Oxford University Press, New York, 2000. MR 1747914
Mostafa Mbekhta, Sur la théorie spectrale locale et limite des nilpotents, Proc. Amer. Math. Soc. 110 (1990), no. 3, 621–631 (French). MR 1004421, DOI 10.1090/S0002-9939-1990-1004421-1
M. Mbekhta and V. Müller, On the axiomatic theory of spectrum. II, Studia Math. 119 (1996), no. 2, 129–147. MR 1391472, DOI 10.4064/sm-119-2-129-147
T. L. Miller and M. M. Neumann, The single-valued extension property for sums and products of commuting operators, Czechoslovak Math. J. 52(127) (2002), no. 3, 635–642. MR 1923267, DOI 10.1023/A:1021788015637
Vladimir Rakočević, Semi-Browder operators and perturbations, Studia Math. 122 (1997), no. 2, 131–137. MR 1432164, DOI 10.4064/sm-122-2-131-137
Florian-Horia Vasilescu, Analytic functional calculus and spectral decompositions, Mathematics and its Applications (East European Series), vol. 1, D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste România, Bucharest, 1982. Translated from the Romanian. MR 690957
Pavla Vrbová, On local spectral properties of operators in Banach spaces, Czechoslovak Math. J. 23(98) (1973), 483–492. MR 322536, DOI 10.21136/CMJ.1973.101189
Sen Zhu and Chun Guang Li, SVEP and compact perturbations, J. Math. Anal. Appl. 380 (2011), no. 1, 69–75. MR 2786184, DOI 10.1016/j.jmaa.2011.02.036