# Scientific papers

#### Generalized Drazin inverse and commuting Riesz perturbations

M. Oudghiri and K. Souilah, *Springer Proceedings in Mathematics and Statistics*, vol. 311, p. 139-145.

In this note, we provide necessary and sufficient conditions for the stability of generalized Drazin invertible operators under commuting Riesz perturbation. We also focus on the commuting perturbation class of meromorphic operators.

#### Complex symmetric operators and additive preservers problem

Z. Amara, M. Oudghiri and K. Souilah, *Advances in Operator Theory*, vol. 5, p. 261-279.

Given a conjugation `C` on a separable complex Hilbert space `H`, a bounded linear operator `T` on `H` is said to be `C`-symmetric if `CTC=T ^{*}`, and is said to be

`C`-skew symmetric if

`CTC=-T`. In this paper, we provide a complete description of all additive maps, on the algebra of all bounded linear operators acting on

^{*}`H`, that preserve

`C`-symmetric operators for every conjugation

`C`. We focus also on the linear maps preserving

`C`-skew symmetric operators.

#### Nonlinear maps preserving Drazin invertible operators of bounded index

M. Oudghiri and K. Souilah, *Quaestiones Mathematicae*.

Given an integer `n≥1`, we provide a complete description of all bijective bicontinuous maps, on the algebra of all bounded linear operators acting on an infinite-dimensional complex or real Banach space, that preserve the difference of Drazin invertible operators of index non-greater than `n` in both directions.

#### Ascent, descent and additive preserving problems

M. Oudghiri and K. Souilah, *Studia Universitatis Babeş-Bolyai Mathematica*, vol. 64, p. 565-580.

Given an integer `n≥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 the set of all bounded operators with ascent (resp. descent) non-greater than n in both directions. In the context of Banach spaces, we consider the additive preserving problem for semi-Fredholm operators with ascent or descent non-greater than
`n`.

#### The perturbation class of algebraic operators and applications

M. Oudghiri and K. Souilah, *Annals of Functional Analysis*, vol. 9, p. 426-434.

In this paper, we completely describe the perturbation class, as well the commuting perturbation class, and the topological interior of the class of all bounded linear algebraic operators. As applications, we also focus on the stability of the essential ascent spectrum and the essential descent spectrum under finite rank perturbations.

#### Linear preservers of the essential ascent and essential descent

M. Oudghiri and K. Souilah, *Rendiconti del Circolo Matematico di Palermo, Second Series*, vol. 67, p. 227-232.

In this paper we characterize all linear surjective maps on the algebra of all bounded linear operators acting on a complex separable infinite-dimensional Hilbert space, preserving operators of finite essential ascent, operators of finite essential descent, or operators of finite essential ascent and finite essential descent in both directions.

#### A generalization of strongly preserver problems of Drazin invertibility

M. Oudghiri and K. Souilah, *Acta Mathematica Vietnamica*, vol. 43, p. 575-583.

Let `Φ` be an additive map between unital complex Banach algebras such that `Φ(1)` is invertible. We show that `Φ` satisfies
`Φ(a ^{D})Φ(b)^{D}=Φ(a)^{D}Φ(b^{D})`
for every Drazin invertible elements

`a,b`if and only if

`Φ(1)`is a Jordan homomorphism and

^{-1}Φ`Φ(1)`commutes with the range of

`Φ`. A similar result is established for group invertible elements, and more explicit forms of such maps are given in the context of the algebra of all bounded linear operators on a complex Banach space.

#### Nonlinear preservers of group invertible operators

M. Oudghiri and K. Souilah, *Asian-European Journal of Mathematics*, vol. 11, p. 13.

Let `B(X)` be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space `X`. We prove that a bijective bicontinuous map `Φ` on `B(X)` preserves the difference of group invertible operators in both directions if and only if
`Φ` is either of the form
`Φ(T)=αATA ^{-1}+Φ(0)` or of the form

`Φ(T)=αBT`where

^{*}B^{-1}+Φ(0)`α`is a non-zero scalar,

`A:X→X`and

`B:X`are two bounded invertible linear or conjugate linear operators.

^{*}→X#### Additive preservers of Drazin invertible operators with bounded index

M. Oudghiri and K. Souilah, *Acta Mathematicae Sinica, English Series*, vol. 33, 1225-1241.

Given an integer `n≥1`, we provide a complete description of all bijective bicontinuous maps, on the algebra of all bounded linear operators acting on an infinite-dimensional complex or real Banach space, that preserve the difference of Drazin invertible operators of index non-greater than `n`in both directions.

#### Additive maps preserving Drazin-invertible operators of index n

M. Mbekhta, M. Oudghiri and K. Souilah, *Banach Journal of Mathematical Analysis*, vol. 11, p. 416-437.

Given an integer `n≥2`. In this article we provide a complete description of all additive surjective maps in the algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space, preserving in both directions the set of Drazin invertible operators of index `n`.

#### Linear preservers of quadratic operators

M. Oudghiri and K. Souilah, *Mediterranean Journal of Mathematics*, vol. 13, p. 4929-4938.

Let `B(H)` be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space `H`. In this paper, we get a compete classification of surjective linear maps on `B(H)` which preserve quadratic operators in both directions. An analogue result in the setting of finite dimensional Banach spaces is given.

#### Non-linear maps preserving singular algebraic operators

M. Oudghiri and K. Souilah, *Proyecciones Journal of Mathematics*, vol. 35, p. 301-316.

Let $\lh$ be the algebra of all bounded linear operators on an infinite-dimensional Hilbert space $H$. We prove that if $\Phi$ is a surjective map on $\lh$ such that $\Phi(I)=I+\Phi(0)$ and for every pair $T,S\in\lh$, the operator $T-S$ is singular algebraic if and only if $\Phi(T)-\Phi(S)$ is singular algebraic, then $\Phi$ is either of the form $\Phi(T)=ATA^{-1}+\Phi(0)$ or the form $\Phi(T)=AT^*A^{-1}+\Phi(0)$ where $A:H\to H$ is an invertible bounded linear, or conjugate linear, operator.

#### Additive maps preserving Drazin invertible operators of index one

M. Mbekhta, M. Oudghiri and K. Souilah, *Mathematical Proceedings of the Royal Irish Academy*, vol. 116A, p. 19-34.

The purpose of the present paper is to completely describe the form of all additive surjective maps in the algebra of all bounded linear operators acting on a complex infinite-dimensional Banach space, preserving in both directions the set of Drazin invertible operators of index one.

#### On additive preservers of certain classes of algebraic operators

K. Souilah, *Extracta Mathematicae*, vol. 30, p. 207-220.

In this article we provide a complete description of all additive surjective unital maps in the algebra of all bounded linear operators acting on an infinite-dimensional Hilbert space, preserving in both directions the set of non-invertible algebraic operators or the set of invertible algebraic operators.