Characterization and stability of essential pseudospectra by measure of polynomially inessential operators

Bilel Elgabeur

doi:10.24193/subbmath.2025.4.11

Authors

Bilel Elgabeur Aix Marseille University, France https://orcid.org/0000-0002-3595-597X

DOI:

https://doi.org/10.24193/subbmath.2025.4.11

Keywords:

Pseudo spectrum, Essential pseudospectra, strict singular operators, polynomially strict singular operators.

Abstract

In this article, we study the essential pseudospectra by measure of polynomially strict singular operators, which is a generalization of the class of strict singular operators. We present some new results in essential pseudospectra for closed linear operators in Banach space with polynomially strict singular operators.
Furthermore, we apply the obtained results to analyze the incidence of some perturbation results on left(resp. right) Weyl essential pseudospectra and left(resp. right) Fredholm essential pseudospectra. In addition, we will describe the essential pseudospectra of a sum of two bounded linear operators. A final application of the obtained results is to characterize the pseudo-left (right)-Fredholm spectra of 2 x 2 block operator matrices.

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Author Biography

Bilel Elgabeur, Aix Marseille University, France

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References

Abramovich, Y. A., Aliprantis, C. D., An Invitation to Operator Theory, American Mathematical Society, 2002.

Abdmouleh, F., Ammar, A., Jeribi, A., A characterization of the pseudo-Browder essential spectra of linear operators and application to a transport equation, J. Comput. Theor. Transp., 44(3)(2015), 141–153.

Abdmouleh, F., Elgabeur, B., Pseudo essential spectra in Banach space and application to operator matrices, Acta Appl. Math., 178(2022), 1–17.

Abdmouleh, F., Elgabeur, B., On the pseudo semi-Browder essential spectra and application to 2 × 2 block operator matrices, Filomat, 37(19)(2023), 6373–6386.

Ammar, A., Boukettaya, B., Jeribi, A., A note on the essential pseudospectra and application, Linear Multilinear Algebra, 64(8)(2016), 1474–1483.

Ammar, A., Jeribi, A., Mahfoudhi, K., Browder essential approximate pseudospectrum and defect pseudospectrum on a Banach space, Extracta Math., 34(1)(2019), 29–40.

Bessaga, C., Pelczynski, A., On strictly singular and strictly cosingular operators, Acta Math.

Charfi, S., Elleuch, A., Walha, I., Spectral theory involving the concept of quasi-compact perturbations, Mediterr. J. Math., 17(2020), No. 32.

Davies, E. B., Spectral Theory and Differential Operators, Cambridge University Press, Cambridge, vol. 42 of Cambridge Studies in Advanced Mathematics, 1995.

Dehici, A., Boussetila, N., Properties of polynomially Riesz operators on some Banach spaces, Lobachevskii J. Math., 32(1)(2011), 39–47.

Elgabeur, B., A characterization of essential pseudospectra involving polynomially compact operators, Filomat, 38(2024), 11675–11691.

Elgabeur, B., A New Approach of Essential Pseudo Spectrum in Banach Space and Application to Transport Equation, Bol. Soc. Paran. Mat., 43(2025), 1–14.

Gohberg, I. C., Markus, A., Feldman, I. A., Normally solvable operators and ideals associated with them, Amer. Math. Soc. Transl., 2(61)(1967), 63–84.

Goldberg, S., Unbounded linear operators: Theory and Applications, McGraw-Hill New York, 1966.

González, M., Onieva, M. O., On Atkinson operators in locally convex spaces, Math. Z., 190(1985), 505-517.

Gustafson, K., Weidmann, J., On the essential spectrum, J. Math. Anal. Appl., 25(6)(1969), 121-127.

Hinrichsen, D., Pritchard, A. J., Robust stability of linear evolution operators on Banach spaces, SIAM J. Control Optim., 32(6)(1994), 1503–1541.

Jeribi, A., Moalla, N., Yengui, S., Some results on perturbation theory of matrix operators, M-essential spectra and application to an example of transport operators, J. Math. Appl., 44(2021).

Kato, T., Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.

Landau, H. J., On Szego's eigenvalue distribution theorem and non-Hermitian kernels, J. Anal. Math., 28(1975), 335-357.

Lindenstrauss, J., Tzafriri, L., Classical Banach Spaces I, Springer-Verlag, Berlin, 1977.

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

Pietsch, A., Operator Ideals, North-Holland, Amsterdam, 1980.

Rakocevic, V., Approximate point spectrum and commuting compact perturbations, Glasg. Math. J., 28(2)(1986), 193–198.

Schmoeger, M., The spectral mapping theorem for the essential approximate point spectrum, Colloq. Math., 74(2)(1997), 167–176.

Schechter, M., Principles of Functional Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 36, American Mathematical Society, Providence, RI, 2002.

Trefethen, L. N., Embree, M., Spectra and Pseudospectra of Linear Operators, Princeton University Press, 2005.

Trefethen, L. N., Pseudospectra of linear operators, SIAM Rev., 39(3)(1997), 383–406.

Varah, J. M., The computation of bounds for the invariant subspaces of a general matrix operator, Ph.D. Thesis, Computer Science Department, Stanford University, 1967.

Zivkovic-Zlatanovic, S. C., Djordjevic, D. S., Harte, R. E., Left-right Browder and left-right Fredholm operators, Integral Equations Operator Theory, 69(2011), 347-363.

Zivkovic-Zlatanovic, S. C., Djordjevic, D. S., Harte, R. E., On left and right Browder operators, J. Korean Math. Soc., 48(5)(2011), 1053–1063.