Ascent, descent and additive preserving problems

Mourad Oudghiri; Khalid Souilah

doi:10.24193/subbmath.2019.4.10

Authors

Mourad Oudghiri Université Mohammed Premier
Khalid Souilah Université Mohammed Premier

DOI:

https://doi.org/10.24193/subbmath.2019.4.10

Keywords:

Linear preserver problems, ascent, descent, semi-Fredholm operators

Abstract

Given an integer $n\geq 1$, we provide a complete description of all additive surjective maps, on the algebra of all bounded linear operators acting on a complex separable infinite-dimensional Hilbert space, preserving in both directions the set of all bounded linear operators with ascent (resp. descent) non-greater than $n$. In the context of Banach spaces, we consider the additive preserving problem for semi-Fredholm operators with ascent or descent non-greater than $n$.

References

B. Aupetit, A Primer on Spectral Theory, Springer-Verlag, New York, 1991.

O. Bel Hadj Fredj, M. Burgos and M. Oudghiri, Ascent and essential ascent spectrum, Studia Math. 187, (2008), 59-73.

M. Burgos, A. Kaidi, M. Mbekhta and M. Oudghiri, The descent spectrum and perturbations, J. Operator Theory {bf 56} (2006), 259-271.

S. Grabiner and J. Zem'anek, Ascent, descent, and ergodic properties of linear operators, J. Operator Theory {bf 48} (2002), 69-81.

A. A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. {bf 66} (1986), 255-261.

M. Kuczma, An introduction to the Theory of Functional Equations and Inequalities, Pa'nstwowe Wydawnictwo Naukowe, Warszawa, 1985.

M. Mbekhta, V. M"uller and M. Oudghiri, Additive preservers of the ascent, descent and related subsets, J. Operator Theory {bf 71} (2014), 63-83.

M. Mbekhta, V. M"uller and M. Oudghiri, On Additive preservers of semi-Browder operators, Rev. Roumaine Math. Pures Appl. {bf 59} (2014), 237-244.

M. Mbekhta and M. Oudghiri, On additive maps preserving certain semi-Fredholm subsets, Linear Multilinear Algebra {bf 61} (2012), 1010-1016.

M. Oudghiri and K. Souilah, Additive preservers of Drazin invertible operators with bounded index, Acta Math. Sin. (Engl. Ser.) {bf 33} (2017), 1225-1241.

V. M"uller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Second edition. Operator Theory: Advances and Applications, {bf 139}. Birkh"auser Verlag, Basel, 2007.

M. Omladiv{c} and P. v{S}emrl, Additive mappings preserving operators of rank one, Linear Algebra Appl. {bf 182} (1993), 239-256.

A. E. Taylor and D. C. Lay,Introduction to Functional Analysis, Wiley, New York-Chichester-Brisbane, 1980.

A. E. Taylor, Theorems on Ascent, Descent, Nullity and Defect of Linear Operators, Math. Ann. {bf 163} (1966), 18-49.