Characterization of the structured pseudospectrum in non-Archimedean Banach spaces

Document Type : Research Article

Author

Department of Mathematics, Sidi Mohamed Ben Abdellah University, Fez, Morocco.

Abstract

In this paper, we demonstrate some results on the pseudospectrum of linear operator pencils on non-Archimedean Banach spaces. In particular, we give a relationship between the Fredholm spectrum of a bounded operator pencil $(A,B)$ and the Fredholm spectrum of the pencil $(A^{-1},B^{-1}).$ Also, we establish a characterization of the essential spectrum of operator pencils on non-Archimedean Banach spaces. Furthermore, we introduce and study the structured pseudospectrum of linear operators on non-Archimedean Banach spaces. We prove that the structured pseudospectra associated with various $\varepsilon$ are nested sets and the intersection of all the structured pseudospectra is the spectrum. We establish a characterization of the structured pseudospectrum of bounded linear operators on non-Archimedean Banach spaces. Finally, we characterize the structured essential pseudospectrum of bounded linear operator pencils on non-Archimedean Banach spaces and we give an illustrative example.

Keywords

Main Subjects


[1] F. Abdmouleh, A. Ammar and A. Jeribi, Stability of the S-Essential Spectra on a Banach Space, Math. Slovaca 63 (2013), no. 2, 299–320.
[2] A. Ammar, A. Bouchekoua and A. Jeribi, Pseudospectra in a Non-Archimedean Banach Space and Essential Pseudospectra in Eω, Filomat 33 (2019), no. 12, 3961–3976.
[3] J. Araujo, C. Perez-Garcia and S. Vega, Preservation of the index of p-adic linear operators under compact perturbations, Compositio Math. 118 (1999), no. 3, 291–303.
[4] A. Blali, A. El Amrani and J. Ettayb, Some spectral sets of linear operator pencils on non-Archimedean Banach spaces, Bull. Transilv. Univ. Braşov Ser. III. Math. Comput. Sci. 2(64) (2022), no. 1, 41–56.
[5] A. Blali, A. El Amrani and J. Ettayb, A note on Pencil of bounded linear operators on non-Archimedean Banach spaces, Methods Funct. Anal. Topology 28 (2022), no. 2, 105–109.
[6] T. Diagana and F. Ramaroson, Non-archimedean Operators Theory, Springer, 2016.
[7] E. B. Davies, Linear Operators and Their Spectra, Cambridge University Press, New York, 2007.
[8] A El Amrani, J Ettayb and A Blali, Pseudospectrum and condition pseudospectrum of non-archimedean matrices, Prime Res. Math. 18 (2022), no. 1, 75–82.
[9] A. El Amrani, A. Blali and J. Ettayb, On Pencil of Bounded Linear Operators on Non-archimedean Banach Spaces, Bol. Soc. Paran. Mat. 42 (2024), 1–10.
[10] J. Ettayb, Pseudospectrum and essential pseudospectrum of bounded linear operator pencils on non-Archimedean Banach spaces, Bol. Soc. Paran. Mat, to appear.
[11] J. Ettayb, Pseudospectrum of non-Archimedean matrix pencils, Bulletin of the Transilvania University of Braşov Series III: Mathematics and Computer Science, in press.
[12] J. Ettayb, Structured pseudospectrum and structured condition pseudospectrum of non-archimedean matrices, arXiv preprint arXiv:2211.10365, 2022.
[13] S. N. Krishnamachari, Linear Operators between Nonarchimedean Banach Spaces, Dissertations, Western Michigan University, Ann Arbor, 1973.
[14] H. R. Henriquez, H. G. Samuel Navarro and J. Aguayo, Closed linear operators between nonarchimedean Banach spaces, Indag. Math. (N.S.) 16 (2005), no. 2, 201–214.
[15] A. Jeribi, Linear operators and their essential pseudospectra, Apple Academic Press, 2018.
[16] A. F. Monna, Analyse non-archimédienne, Springer, Berlin, 1970.
[17] C. Perez-Garcia and S. Vega, Perturbation theory of p-adic Fredholm and semi-Fredholm operators, Indag. Math. (N.S.) 15 (2004), no. 1, 115–128.
[18] A. C. M. van Rooij, Non-Archimedean functional analysis, Monographs and Textbooks in Pure and Applied Math. Marcel Dekker, Inc., New York, 1978.
[19] L. N. Trefethen and M. Embree, Spectra and Pseudospectra, The behavior of nonnormal matrices and operators, Princeton University Press, Princeton, 2005.