Abstract
In this paper, we consider \({\mathcal A}\)-Fredholm and semi-\({\mathcal A}\)-Fredholm operators on Hilbert C*-modules over a W*-algebra \({\mathcal A}\) defined in [3] and [9]. Using the assumption that \({\mathcal A}\) is a W*-algebra (rather than an arbitrary C*-algebra), we obtain a generalization of Schechter—Lebow characterization of semi-Fredholm operators and a generalization of the “punctured neighborhood” theorem, as well as some other results generalizing their classical counterparts. We consider both adjointable and nonadjointable semi-Fredholm operators over W*-algebras. Moreover, we also work with general bounded adjointable operators with closed ranges over C*-algebras and prove a generalization of a Bouldin result for Hilbert spaces to Hilbert C*-modules.
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References
R. Bouldin, “The Product of Operators with Closed Range,” Tōhoku Math. J. 25, 359–363 (1973).
M. Frank and E. V. Troitsky, “Lefschetz Numbers and Geometry of Operators in W*-Modules,” Funktsional. Anal. i Priloshen. 30(4), 45–57 (1996).
S. Ivković, “Semi-Fredholm Theory on Hilbert C*-Modules,” Banach J. Math. Anal., to appear (2019), 13 (4), arXiv: https://arxiv.org/abs/1906.03319.
A. A. Irmatov and A. S. Mishchenko, “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).
E. C. Lance, “On Nuclear C*-Algebras,” J. Funct. Anal. 12, 157–176 (1973).
D. Lay, “Spectral Analysis Using Ascent, Descent, Nullity, and Defect,” Math. Ann. 184, 197–214 (1970).
H. Lin, “Injective Hilbert C*-Modules,” Pacific J. Math. 154, 133–164 (1992).
A. Lebow and M. Schechter, “Semigroups of Operators and Measures of Non-Compactness,” J. Funct. Anal. 7, 1–26 (1971).
A. S. Mishchenko and A. T. Fomenko, “The Index of Eliptic Operators over C*-Algebras,” Izv. Akad. Nauk SSSR Ser. Mat. 43, 831–859 (1979)
A. S. Mishchenko and A. T. Fomenko, English transl., Math. USSR-Izv. 15, 87–112 (1980).
V. M. Manuilov and E. V. Troitsky, “Hilbert C*-Modules,” Translations of Mathematical Monographs 226 (Amer. Math. Soc., Providence, RI, 2005).
G. Nikaido, “Remarks on the Lower Bound of a Linear Operator,” Proc. Japan Acad. Ser. A Math. Sci. 56(7), 321–323 (1980).
K. Sharifi, “The Product of Operators with Closed Range in Hilbert C*-Modules,” Linear Algebra Appl. 435, 1122–1130 (2011).
M. Schechter, “Quantities Related to Strictly Singular Operators,” Indiana Univ. Math. J. 21(11), 1061–1071 (1972).
M. O. Searooid, “The Continuity of the Semi-Fredholm Index,” IMS Bull. 29, 13–18 (1992).
N. E. Wegge-Olsen, K-Theory and C*-Algebras (Oxford Univ. Press, Oxford, 1993).
S. Živković-Zlatanović, V. Rakočević, and D. S. Djordjević, Fredholm Theory (University of Niš Faculty of Sciences and Mathematics, Niš, to appear, 2019).
Acknowledgment
I am especially grateful to my research supervisor Professor Vladimir M. Manuilov for carefully reading my paper and for inspiring comments and suggestions that led to an improved presentation of the text. I am also grateful to Professor Dragan S. Djordjevic for suggesting the research topic of the paper and for introducing the relevant reference books to me.
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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
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DOI: https://doi.org/10.1134/S1061920820010057