Document Type : Research Paper
Author
Department of Mathematics and Computer Science, Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, B.P. 1796 Atlas, F\`es, Morocco.
Abstract
In this paper, we define the notions of semi-regular operator, analytical core, surjectivity modulus and the injectivity modulus of bounded linear operators on non-Archimedean Banach spaces over $\mathbb{K}.$ We give a necessary and sufficient condition on the range of bounded linear operators to be closed. Moreover, many results are proved.
Keywords
- Non-Archimedean Banach spaces
- Bounded linear operators
- Range of bounded linear operators
- Reduced minimum modulus
Main Subjects
[1] P. Aiena, Fredholm and local spectral theory, with applications to multipliers, Kluwer Academic Publishers, Dordrecht, 2004.
[2] J. Araujo, C. Perez-Garcia and S. Vega, Preservation of the index of $p$-adic linear operators under compact perturbations, Compos. Math., 118(3) (1999), pp. 291-303.
[3] T. Diagana and F. Ramaroson, Non-archimedean Operator Theory, SpringerBriefs in Mathematics, Springer, Cham, 2016.
[4] B. Diarra, An operator on some ultrametric Hilbert spaces, J. Anal., 6 (1998), pp. 55-74.
[5] P.J.T. Filippi, O. Lagarrigue and P.O. Mattei, Perturbation method for sound radiation by a vibrating plate in a light fluid: comparison with the exact solution, Journal of sound and vibration, 177 (2) (1994), pp. 259-275.
[6] L. Gruson, Théorie de Fredholm $p$-adique, Bull. Soc. Math. France, 94 (1966), pp. 67–95.
[7] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966.
[8] J. Killingbeck, Quantum-mechanical perturbation theory, Rep. Prog. Phys., 40(9) (1977), pp. 963.
[9] T. Kiyosawa, Perturbations of $p$-adic linear operators, Rocky Mountain J. Math., 34(3) (2004), pp. 991-1004.
[10] C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over Non-archimedean Valued Fields, Cambridge University Press, Cambridge, 2010.
[11] P. Robba, On the index of $p$-adic differential operators $I$, Ann. of Math., (2) 101 (1975), pp. 280–316.
[12] P. Robba and G. Christol, Equations différentielles $p$-adiques: Applications aux sommes exponentielles, Actualites Mathe., Hermann, Paris, 1994.
[13] A.C.M. van Rooij, Non-Archimedean functional analysis, Marcel Dekker Inc., New York, 1978.
[14] P. Saphar, Contribution àl'étude des applications linéaires dans un espace de Banach, Bull. Soc. Math. France, 92 (1964), pp. 363-384.
[15] W.H. Schikhof, On $p$-adic Compact Operators, Report 8911, Department of Mathematics, Catholic University, Nijmegen, The Netherlands, (1989), pp. 1–28.
[16] J.P. Serre, Endomorphismes complètement continus des espaces de Banach $p$-adiques, Publ. Math. Inst. Hautes Etudes Sci., 12 (1962), pp. 69–85.
)
