Characterization of essential spectra by quasi-compact perturbations


  • Faiçal Abdmouleh University of Sfax, Higher Institute of Industrial Management of Sfax, Route de Tunis km 10.5, Technopole de Sfax, BP 1164-3021, Sfax, Tunisia
  • Hamadi Chaâben University of Sfax, Faculty of Sciences of Sfax, Route de la Soukra km 4, BP 1171-3000, Sfax, Tunisia
  • Ines Walha University of Sfax, Faculty of Sciences of Sfax, Department of Mathematics, Route de la Soukra km 4, BP 1171-3000, Sfax, Tunisia



We are interested in the concept of quasi-compact operators allowing us to provide some advances on the theory of operators acting in Banach spaces. More precisely, our main objective is to exhibit the importance of the use of this notion to outline a new approach in the analysis of the stability problems of upper and lower semi-Fredholm, upper and lower semi-Weyl, and upper and lower semi-Browder operators, and to provide a fine description and characterization of some Browder's essential spectra involving this kind of operators.


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F. Abdmouleh and I. Walha, Characterization and stability of the essential spectrum based on measures of polynomially non-strict singularity operators, Indag. Math. (N.S.) 26 no. 3 (2015), 455–467.  DOI  MR  Zbl

F. Abdmouleh and I. Walha, Measure of non strict singularity of Schechter essential spectrum of two bounded operators and application, Bull. Iranian Math. Soc. 43 no. 5 (2017), 1543–1558.  MR  Zbl

P. Aiena, Fredholm and Local Spectral Theory, With Applications to Multipliers, Kluwer, Dordrecht, 2004.  MR  Zbl

A. Brunel and D. Revuz, Quelques applications probabilistes de la quasi-compacité, Ann. Inst. H. Poincaré Sect. B (N.S.) 10 (1974), 301–337.  MR  Zbl

S. R. Caradus, Operators with finite ascent and descent, Pacific J. Math. 18 (1966), 437–449.  MR  Zbl Available at

S. Charfi and I. Walha, On relative essential spectra of block operator matrices and application, Bull. Korean Math. Soc. 53 no. 3 (2016), 681–698.  DOI  MR  Zbl

L. Chen and W. Su, A note on Weyl-type theorems and restrictions, Ann. Funct. Anal. 8 no. 2 (2017), 190–198.  DOI  MR  Zbl

M. A. Golʹdman and S. N. Kračkovskiĭ, Behavior of the space of zeros with a finite-dimensional salient on the Riesz kernel under perturbations of the operator, Dokl. Akad. Nauk SSSR 221 no. 13 (1975), 532–534, English translation in Sov. Math., Dokl. 16 (1975), 370–373.  MR  Zbl

S. Grabiner, Ascent, descent and compact perturbations, Proc. Amer. Math. Soc. 71 no. 1 (1978), 79–80.  DOI  MR  Zbl

R. Harte, Invertibility and Singularity for Bounded Linear Operators, Monographs and Textbooks in Pure and Applied Mathematics 109, Marcel Dekker, New York, 1988.  MR  Zbl

H. Hennion and L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics 1766, Springer-Verlag, Berlin, 2001.  DOI  MR  Zbl

M. A. Kaashoek, Ascent, descent, nullity and defect: A note on a paper by A. E. Taylor, Math. Ann. 172 (1967), 105–115.  DOI  MR  Zbl

M. A. Kaashoek and D. C. Lay, Ascent, descent, and commuting perturbations, Trans. Amer. Math. Soc. 169 (1972), 35–47.  DOI  MR  Zbl

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften 132, Springer-Verlag, New York, 1966.  MR  Zbl

U. A. Koumba, On Riesz Operators, Ph.D. thesis, University of Johannesburg, 2014. Available at

N. Kryloff and N. Bogoliouboff, Les propriétés ergodiques des suites des probabilités en chaîne, C. R. Acad. Sci., Paris 204 (1937), 1454–1456.  Zbl

M. Lin, On quasi-compact Markov operators, Ann. Probability 2 (1974), 464–475.  DOI  MR  Zbl

J. Martínez and J. M. Mazón, Quasi-compactness of dominated positive operators and $C_0$-semigroups, Math. Z. 207 no. 1 (1991), 109–120.  DOI  MR  Zbl

V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Operator Theory: Advances and Applications 139, Birkhäuser Verlag, Basel, 2003.  DOI  MR  Zbl

V. Rakočević, Approximate point spectrum and commuting compact perturbations, Glasgow Math. J. 28 no. 2 (1986), 193–198.  DOI  MR  Zbl

V. Rakočević, Semi-Browder operators and perturbations, Studia Math. 122 no. 2 (1997), 131–137.  DOI  MR  Zbl

M. Schechter, Principles of Functional Analysis, second ed., Graduate Studies in Mathematics 36, American Mathematical Society, Providence, RI, 2002.  DOI  MR  Zbl

R. F. Taylor, Invariant Subspaces in Hilbert and Normed Spaces, Ph.D. thesis, California Institute of Technology, 1968.  DOI

T. T. West, A Riesz-Schauder theorem for semi-Fredholm operators, Proc. Roy. Irish Acad. Sect. A 87 no. 2 (1987), 137–146.  MR  Zbl