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r-FREDHOLM THEORY IN BANACH ALGEBRAS

Part of: General theory of linear operators Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  25 September 2018

RONALDA BENJAMIN ,
NIELS JAKOB LAUSTSEN  and
SONJA MOUTON
RONALDA BENJAMIN*
Affiliation:
Department of Mathematical Sciences, Private Bag X1, Stellenbosch University, Matieland 7602, South Africa e-mail: ronalda@sun.ac.za
NIELS JAKOB LAUSTSEN*
Affiliation:
Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom e-mail: n.laustsen@lancaster.ac.uk
SONJA MOUTON*
Affiliation:
Department of Mathematical Sciences, Private Bag X1, Stellenbosch University, Matieland 7602, South Africa e-mail: smo@sun.ac.za
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Abstract

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Harte (1982, Math. Z.179, 431–436) initiated the study of Fredholm theory relative to a unital homomorphism T: A → B between unital Banach algebras A and B based on the following notions: an element a ∈ A is called Fredholm if 0 is not in the spectrum of Ta, while a is Weyl (Browder) if there exist (commuting) elements b and c in A with a = b + c such that 0 is not in the spectrum of b and c is in the null space of T. We introduce and investigate the concepts of r-Fredholm, r-Weyl and r-Browder elements, where 0 in these definitions is replaced by the spectral radii of a and b, respectively.

MSC classification

Primary: 46H30: Functional calculus in topological algebras
Secondary: 47A10: Spectrum, resolvent 47A53: (Semi-) Fredholm operators; index theories
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 3 , September 2019 , pp. 615 - 627
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

Alekhno, E. A., Some properties of essential spectra of a positive operator, Positivity 11 (2007), 375–386.CrossRefGoogle Scholar
Alekhno, E. A., Some properties of essential spectra of a positive operator, II, Positivity 13 (2009), 3–20.CrossRefGoogle Scholar
Aupetit, B., A primer on spectral theory (Springer-Verlag, New York, 1991).CrossRefGoogle Scholar
Benjamin, R. and Mouton, S., Fredholm theory in ordered Banach algebras, Quaest. Math. 39 (2016), 643–664.CrossRefGoogle Scholar
Benjamin, R. and Mouton, S., The upper Browder spectrum property, Positivity 21 (2017), 575–592.CrossRefGoogle Scholar
Caradus, S. R., Pfaffenberger, W. E. and Yood, B., Calkin algebras and algebras of operators on Banach spaces (Marcel Dekker, Inc., New York, 1974).Google Scholar
Conway, J. B., Functions of one complex variable I, 2nd edition (Springer, New York, 1978).CrossRefGoogle Scholar
Grobler, J. J. and Raubenheimer, H., Spectral properties of elements in different Banach algebras, Glasgow Math. J. 33 (1991), 11–20.CrossRefGoogle Scholar
Harte, R. E., The exponential spectrum in Banach algebras, Proc. Amer. Math. Soc. 58 (1976), 114–118.CrossRefGoogle Scholar
Harte, R. E., Fredholm theory relative to a Banach algebra homomorphism, Math. Z. 179 (1982), 431–436.CrossRefGoogle Scholar
Harte, R. E., Fredholm, Weyl and Browder theory, Proc. Roy. Irish Acad. Sect. A 85 (1985), 151–176.Google Scholar
Harte, R. E., Fredholm, Weyl and Browder theory II, Proc. Roy. Irish Acad. Sect. A 91 (1991), 79–88.Google Scholar
Koliha, J. J., A generalized Drazin inverse, Glasgow Math. J. 38 (1996), 367–381.CrossRefGoogle Scholar
Lebow, A. and Schechter, M., Semigroups of operators and measures of noncompactness, J. Funct. Anal. 7 (1971), 1–26.CrossRefGoogle Scholar
Lindeboom, L. and Raubenheimer, H., On regularities and Fredholm theory, Czechoslovak Math. J. 52 (2002), 565–574.CrossRefGoogle Scholar
Mathieu, M. and Schick, G. J., First results on spectrally bounded operators, Studia Math. 152 (2002), 187–199.CrossRefGoogle Scholar
du, H. Mouton, T., Mouton, S. and Raubenheimer, H., Ruston elements and Fredholm theory relative to arbitrary homomorphisms, Quaest. Math. 34 (2011), 341–359.Google Scholar
du, H. Mouton, T. and Raubenheimer, H., Fredholm theory relative to two Banach algebra homomorphisms, Quaest. Math. 14 (1991), 371–382.Google Scholar
du, H. Mouton, T. and Raubenheimer, H., More on Fredholm theory relative to a Banach algebra homomorphism, Proc. Roy. Irish Acad. Sect. A 93 (1993), 17–25.Google Scholar
Mouton, S., A spectral problem in ordered Banach algebras, Bull. Austral. Math. Soc. 67 (2003), 131–144.CrossRefGoogle Scholar
Willard, S., General topology (Addison-Wesley, Reading, Massachusetts, 1970).Google Scholar