Abstract
In this paper, we develop some generalized Fredholm perturbation results for bounded linear operators acting on non-reflexive Banach spaces satisfying certain conditions. Furthermore, we investigate the stability of the generalized Schechter S-essential spectrum by means of the so-called generalized Fredholm index. This concept is introduced as a semigroup homomorphism satisfying certain properties. Some generalized Fredholm results for block operator matrices acting on a non-reflexive product of two Banach spaces are also developed.
References
[1] P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic, Dordrecht, 2004. Search in Google Scholar
[2] F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Grad. Texts in Math. 233, Springer, New York, 2006. Search in Google Scholar
[3] K. Astala, On measures of noncompactness and ideal variations in Banach spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. Diss. 29 (1980), 1–42. Search in Google Scholar
[4] A. Azzouz, M. Beghdadi and B. Krichen, g-Riesz operators and their spectral properties, Adv. Oper. Theory 7 (2022), no. 2, Paper No. 18. 10.1007/s43036-021-00179-6Search in Google Scholar
[5] A. Azzouz, M. Beghdadi and B. Krichen, Generalized relative essential spectra, Filomat 36 (2022), no. 8, 2657–2673. 10.2298/FIL2208657ASearch in Google Scholar
[6] J. Banaś and J. Rivero, On measures of weak noncompactness, Ann. Mat. Pura Appl. (4) 151 (1988), 213–224. 10.1007/BF01762795Search in Google Scholar
[7] F. S. De Blasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N. S.) 21(69) (1977), no. 3–4, 259–262. Search in Google Scholar
[8] N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, Pure Appl. Math. 7, Interscience, New York, 1958. Search in Google Scholar
[9] G. Emmanuele, Measure of weak noncompactness and fixed point theorems, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N. S.) 25(73) (1981), no. 4, 353–358. Search in Google Scholar
[10] I. C. Gohberg, A. S. Markus and I. A. Feldman, Normally solvable operators and ideals associated with them, Amer. Math. Soc. Transl. Ser. 2 (1967), no. 61, 63–84. 10.1090/trans2/061/03Search in Google Scholar
[11] S. Goldberg, Unbounded Linear Operators: Theory and Applications, McGraw-Hill, New York, 1966. Search in Google Scholar
[12] M. González and A. Martínez-Abejón, Tauberian Operators, OOper. Theory Adv. Appl. 194, Birkhäuser, Basel, 2010. 10.1007/978-3-7643-8998-7Search in Google Scholar
[13] B. Gramsch and D. Lay, Spectral mapping theorems for essential spectra, Math. Ann. 192 (1971), 17–32. 10.1007/BF02052728Search in Google Scholar
[14] A. Jeribi, Spectral Theory and Applications of Linear Operators and Block Operator Matrices, Springer, Cham, 2015. 10.1007/978-3-319-17566-9Search in Google Scholar
[15] A. Jeribi and B. Krichen, Nonlinear Functional Analysis in Banach Spaces and Banach Algebras. Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications, Monogr. Res. Notes Math., CRC Press, Boca Raton, 2016, 10.1201/b18790Search in Google Scholar
[16] A. Jeribi and M. Mnif, Fredholm operators, essential spectra and application to transport equations, Acta Appl. Math. 89 (2005), no. 1–3, 155–176. 10.1007/s10440-005-9005-2Search in Google Scholar
[17] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss. 132, Springer, New York, 1966. 10.1007/978-3-642-53393-8Search in Google Scholar
[18] B. Krichen and D. O’Regan, Weakly demicompact linear operators and axiomatic measures of weak noncompactness, Math. Slovaca 69 (2019), no. 6, 1403–1412. 10.1515/ms-2017-0317Search in Google Scholar
[19] K. Latrach and A. Dehici, Fredholm, semi-Fredholm perturbations, and essential spectra, J. Math. Anal. Appl. 259 (2001), no. 1, 277–301. 10.1006/jmaa.2001.7501Search in Google Scholar
[20] K. Latrach and A. Jeribi, Some results on Fredholm operators, essential spectra, and application, J. Math. Anal. Appl. 225 (1998), no. 2, 461–485. 10.1006/jmaa.1998.6038Search in Google Scholar
[21] A. Peł czyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. 10.4064/sm-19-2-209-228Search in Google Scholar
[22] M. Schechter, Principles of Functional Analysis, Academic Press, New York, 1971. Search in Google Scholar
[23] C. Tretter, Spectral Theory of Block Operator Matrices and Applications, Imperial College, London, 2008. 10.1142/p493Search in Google Scholar
[24] K. W. Yang, The generalized Fredholm operators, Trans. Amer. Math. Soc. 216 (1976), 313–326. 10.1090/S0002-9947-1976-0423114-XSearch in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The Wigner transformation associated with the Hankel multidimensional operator
- Index for generalized Fredholm operators and generalized perturbation theory
- One-radius theorem for harmonic tempered distributions
- Optimal convergence factors for general Fourier coefficients
- Natural transformations for quasigroupoids
- Numerical solution for a nonlinear diffusion model with source terms
- Multiplicative (generalized)-reverse derivations in rings and Banach algebras
- Variational approach of p-Laplacian impulsive differential equations with periodic conditions
- Solvable groups with four conjugacy classes outside a normal subgroup
- Quadratic-phase orthonormal wavelets
- Uncertainty inequality on weighted Hardy spaces
- Faithful representations of the Galilean Lie algebra in two spatial dimensions
- Weighted composition operators from Dirichlet–Zygmund-type spaces into Stević-type spaces
Articles in the same Issue
- Frontmatter
- The Wigner transformation associated with the Hankel multidimensional operator
- Index for generalized Fredholm operators and generalized perturbation theory
- One-radius theorem for harmonic tempered distributions
- Optimal convergence factors for general Fourier coefficients
- Natural transformations for quasigroupoids
- Numerical solution for a nonlinear diffusion model with source terms
- Multiplicative (generalized)-reverse derivations in rings and Banach algebras
- Variational approach of p-Laplacian impulsive differential equations with periodic conditions
- Solvable groups with four conjugacy classes outside a normal subgroup
- Quadratic-phase orthonormal wavelets
- Uncertainty inequality on weighted Hardy spaces
- Faithful representations of the Galilean Lie algebra in two spatial dimensions
- Weighted composition operators from Dirichlet–Zygmund-type spaces into Stević-type spaces