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On the operators which are invertible modulo an operator ideal

Published online by Cambridge University Press:  17 April 2009

Pietro Aiena ,
Manuel González  and
Antonio Martínez-Abejón
Pietro Aiena
Affiliation:
Dipartimento di Matematica ed Applicazioni, Università di Palermo, Viale delle Scienze, I-90128 Palermo, Italy
Manuel González
Affiliation:
Departamento de Matemáticas, Universidad de Oviedo, E-33007 Oviedo, Spain
Antonio Martínez-Abejón
Affiliation:
Departamento de Matemáticas, Universidad de Cantabria, E-39071 Santander, Spain
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We study the semigroups l and r of left and right invertible operators modulo an operator ideal , respectively. We show that these semigroups allow us to obtain useful characterisations of the radical rad of  For example, rad; is the perturbation class for l and r.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 2 , October 2001 , pp. 233 - 243
Copyright
Copyright © Australian Mathematical Society 2001

References

[l]Aiena, P. and González, M., ‘Essentially incomparable Banach spaces and Fredholm theory’, Proc. Roy. Irish Acad. Sect. A 93 (1993), 49–59.Google Scholar
[2]Aiena, P., González, M., and Martinez-Abejón, A., ‘Operator semigroups in Banach space theory’, Boll. Un. Mat. ltal. Sez. B Artic. Ric. Mat. (8) 4 (2001), 157–205.Google Scholar
[3]Atkinson, F., ‘Relatively regular operators’, Acta Sci. Math. (Szeged) 15 (1953), 38–56.Google Scholar
[4]Astala, K. and Tylli, H.-O., ‘Seminorms related to weak compactness and to tauberian operators’, Math. Proc. Cambridge Philos. Soc. 107 (1990), 365–375.CrossRefGoogle Scholar
[5]Bessaga, C. and Pelczyński, A., ‘On basis and unconditional convergence of series in Banach spaces’, Studia Math. 17 (1958), 151–164.CrossRefGoogle Scholar
[6]Diestel, J., ‘A survey of results related to the Dunford-Pettis property’, Contemp. Math. 2 (1980), 15–60.CrossRefGoogle Scholar
[7]González, M., ‘On essentially incomparable Banach spaces’, Math. Z. 215 (1994). 621–629.CrossRefGoogle Scholar
[8]González, M. and Martínez-Abejón, A., ‘Lifting unconditionally converging series and semigroups of operators’, Bull. Austral. Math. Soc. 57 (1998), 135–146.CrossRefGoogle Scholar
[9]González, M., Saksman, E. and Tylli, H.-O., ‘Representing non-weakly compact operators’, Studia Math. 113 (1995), 265–282.CrossRefGoogle Scholar
[10]Kalton, N. and Wilansky, A., ‘Tauberian operators in Banach spaces’, Proc. Amer. Math. Soc. 57 (1976), 251–255.CrossRefGoogle Scholar
[11]Kleinecke, D., ‘Almost-finite, compact, and inessential operators’, Proc. Amer. Math. Soc. 14 (1963), 863–868.CrossRefGoogle Scholar
[12]Lebow, A. and Schechter, M., ‘Semigroups of operators and measures of noncompactness’, J. Funct. Anal. 7 (1971), 1–26.CrossRefGoogle Scholar
[13]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I. Sequence spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1977).Google Scholar
[14]Pietsch, A., ‘Inessential operators in Banach spaces’, Integral Equations Operator Theory 1 (1978), 589–591.CrossRefGoogle Scholar
[15]Pietsch, A., Operator ideals (North-Holland, Amsterdam, New York, Oxford, 1980).Google Scholar
[16]Przeworska-Rolewicz, D. and Rolewicz, S., Equations in linear spaces (P.W.N., Warszawa, 1968).Google Scholar
[17]Tacon, D.G., ‘Generalized semi-Fredholm transformations’, J. Austral. Math. Soc. Ser. A 34. (1983), 60–70.CrossRefGoogle Scholar
[18]Yang, K.W., ‘Operators invertible modulo the weakly compact operators’, Pacific J. Math. 71 (1977), 559–564.CrossRefGoogle Scholar
[19]Yood, B., ‘Difference algebras of linear transformations on a Banach space’, Pacific J. Math. 4 (1954), 615–636.CrossRefGoogle Scholar