Abstract
Starting from the definition of \({\mathcal {A}}\)-Fredholm and semi-\({\mathcal {A}}\)-Fredholm operator on the standard module over a unital \(C^{*}\) algebra \({\mathcal {A}}\), introduced in Ivković (Banach J Math Anal 13(4):989–1016, 2019) and Mishchenko and Fomenko (Izv Akad Nauk SSSR Ser Mat 43:831–859, 1979), we construct various generalizations of these operators and obtain several results as an analogue or a generalization of the results in Berkani and Sarih (Glasg Math J 43(3):457–465, 2001. https://doi.org/10.1017/S0017089501030075), Berkani (Proc Am Math Soc 130(6):1717–1723, 2001), Djordjević (Proc Am Math Soc 130(1):81–84, 2001) and Yang (Trans Am Math Soc 216:313–326, 1976). Moreover, we also study non-adjointable semi-\({\mathcal {A}}\)-Fredholm operators as a natural continuation of the work in Irmatov and Mishchenko (J K-Theory 2:329–351, 2008. https://doi.org/10.1017/is008004001jkt034) on non-adjointable \({\mathcal {A}}\)-Fredholm operators and obtain an analogue or a generalization in this setting of the results in Ivković (Banach J Math Anal 13(4):989–1016, 2019; Ann Funct Anal, 2020. https://doi.org/10.1007/43034-019-00034-z).
Similar content being viewed by others
References
Berkani, M., Sarih, M.: On semi B-Fredholm operators. Glasg. Math. J. 43(3), 457–465 (2001). https://doi.org/10.1017/S0017089501030075
Berkani, M.: On semi B-Fredholm operators. Proc. Am. Math. Soc. 130(6), 1717–1723 (2001)
Djordjević, D.S.: Perturbations of spectra of operator matrices. J. Oper. Theory 48, 467–486 (2002)
Djordjević, D.S.: On generalized Weyl operators. Proc. Am. Math. Soc. 130(1), 81–84 (2001)
Harte, R.E.: The ghost of an index theorem. Proc. Am. Math. Soc. 106, 1031–1033 (1989)
Ivković, S.: Semi-Fredholm theory on Hilbert C*-modules. Banach J. Math. Anal. 13(4), 989–1016 (2019)
Ivković, S.: On compressions and generalized spectra of operators over \(C^*\)-algebras. Ann. Funct. Anal. (2020). https://doi.org/10.1007/43034-019-00034-z
Ivković, S.: On operators with closed range and semi-Fredholm operators over W*-algebras. Russ. J. Math. Phys. 27, 48–60 (2020). https://doi.org/10.1134/S1061920820010057
Irmatov, A.A., Mishchenko, A.S.: On compact and Fredholm operators over C*-algebras and a new topology in the space of compact operators. J. K-Theory 2, 329–351 (2008). https://doi.org/10.1017/is008004001jkt034
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976)
Mishchenko, A.S., Fomenko, A.T.: The index of eliptic operators over C*-algebras. Izv. Akad. Nauk SSSR Ser. Mat. 43, 831–859 (1979); English transl., Math. USSR-Izv.15, 87–112 (1980)
Manuilov, V.M., Troitsky, E.V.: Hilbert C*-modules. In: Krasnosel’skiǐ, M.A. (ed.) Translations of Mathematical Monographs, vol. 226. American Mathematical Society, Providence (2005)
Yang, K.W.: The generalized Fredholm operators. Trans. Am. Math. Soc. 216, 313–326 (1976)
Acknowledgements
I am especially grateful to my supervisors Professor Vladimir M. Manuilov and Professor Camillo Trapani for reading my paper and for giving me comments that led to the improved presentation of the paper. Also, I am grateful to Professor Dragan S. Djordjevic for suggesting the research topic of this paper and for introducing to me the relevant reference books. Finally, I am grateful to the Referee for useful comments that led to the improved version of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Daniel Aron Alpay.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Aurelian Gheondea and Serap Oztop.
Rights and permissions
About this article
Cite this article
Ivković, S. On Various Generalizations of Semi-\({\mathcal {A}}\)-Fredholm Operators. Complex Anal. Oper. Theory 14, 41 (2020). https://doi.org/10.1007/s11785-020-00995-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11785-020-00995-3
Keywords
- Generalized \({\mathcal {A}}\)-Fredholm operator
- Generalized \({\mathcal {A}}\)-Weyl operator
- Semi-\({\mathcal {A}}\)-B-Fredholm operator
- Non-adjointable semi-\({\mathcal {A}}\)-Fredholm operator