arXiv subjects

Fadil Chabbabi

Publications and source records attributed to Fadil Chabbabi.

9 recordsLinked to original sources

arXiv2026

Pseudo-Fredholm operators form a regularity on Hilbert spaces

Let $H$ be a complex Hilbert space. We prove that the class of pseudo-Fredholm operators on $H$ is a regularity in the sense of Kordula and Müller. We also prove that the classes of left and right generalized Drazin invertible operators are regularities on $H$. Consequently, the generalized Kato spectrum and the corresponding generalized Drazin spectra satisfy the spectral mapping theorem. We construct an operator $T$ on a separable complex Banach space such that $T^2$ is right generalized Drazin invertible whereas $T$ is not pseudo-Fredholm. In particular, the power axiom need not hold for pseudo-Fredholm operators on Banach spaces. Finally, we show that the accumulation sets of the approximate point and surjectivity spectra define regularities on Banach spaces, yielding another proof of the regularity of generalized Drazin invertible operators in $\B(X)$.

math.FA↗

arXiv2026

On Mbekhta's conjecture and pseudo-Fredholm operators on Banach and Hilbert spaces

We give negative answers to Mbekhta's conjecture \cite[p.~261]{Mbekhta} and to a related conjecture on pseudo-B-Fredholm operators \cite[p.~202]{AZO}. A sufficient condition for pseudo-Fredholmness on Banach spaces is established. It yields a characterization on Hilbert spaces in terms of the closedness of $\cc(T)$, $\cc(T)+\qn(T)$ and $\Ran(T)+\qn(T)$, as well as characterizations of pseudo-B-Fredholm and pseudo-B-Weyl operators on Banach spaces. A counterexample on a separable Banach space shows that the three closedness conditions do not characterize pseudo-Fredholmness in general. We also construct two operators having the same analytic core, quasinilpotent part, and kernels and ranges of all powers, although only one is pseudo-Fredholm. Moreover, a semi-B-Fredholm operator on a Banach space need not be a direct sum of a semi-Fredholm operator and a nilpotent operator, answering the question raised in \cite[p.~113]{AznayKato2023}. Finally, the three closedness conditions in the Hilbert-space characterization are shown to be independent.

math.FA↗

arXiv2026

Surjective maps preserving quasi-nilpotence of products of operators

Let $X$ be an infinite-dimensional complex Banach space and let $\mathcal B(X)$ the full algebra of all bounded linear operator on $X$. In this paper, we give a complete characterization of surjective maps $ϕ:\mathcal B(X)\to\mathcal B(X)$ satisfying \[ r(AB)=0\quad\Longleftrightarrow\quad r(ϕ(A)ϕ(B))=0 \qquad(A,B\in\mathcal B(X)). \] Without assuming additivity, injectivity, or continuity, we show that there exists a function $γ: B(X)\mapsto \C \setminus\{0\}$ such that the map $ϕ$ takes one of the following forms: 1. there exists a bounded linear or conjugate-linear, invertible operator $S:X\to X$ such that : \label{eq:direct} ϕ(A)=γ(A)SAS^{-1},\qquad \text{ for all } \quad A\in B(X), 2. the space $X$ is reflexive and there exists a bounded linear or conjugate-linear, invertible operator $S:X^*\to X$ such that \label{eq:dual} ϕ(A)=γ(A)SA^*S^{-1},\qquad \text{ for all } \quad A\in B(X).

math.FA↗

arXiv2026

Algebraic Direct Sums and Generalized Drazin Invertibility

Let $T$ be a bounded linear operator on a complex Banach space $X$. We prove that the algebraic decomposition \[ X=\K(T)\oplus \Hq(T) \] forces both summands to be closed and hence characterizes generalized Drazin invertibility. This improves the classical characterizations of Mbekhta~\resultcite{Theorem~1.6}{MbekhtaKato}, Schmoeger~\resultcite{Theorem~4}{Schmoeger}, and Dajić and Koliha~\resultcite{Theorem~6.7}{DajicKoliha}, by removing their closedness assumptions. We also show that $T$ is generalized Drazin invertible if and only if \[ \dim\bigl(\K(T)\cap \Hq(T)\bigr)<\infty \quad\text{and}\quad \codim\bigl(\K(T)+\Hq(T)\bigr)<\infty. \] For ordinary Drazin invertibility, we obtain analogous finite-defect characterizations involving the hyperkernel and the hyperrange. We also give a characterization using pointwise nilpotence on an invariant summand. Finally, we consider arbitrary invariant algebraic decompositions and give a counterexample when the inverse of the restriction to one summand is not assumed bounded.

math.FA↗

arXiv2022

On the image of the mean transform

Let $B(H)$ be the algebra of all bounded operators on a Hilbert space $H$. Let $T=V|T|$ be the polar decomposition of an operator $T\in B(H)$. The mean transform of $T$ is defined by $M(T)=\frac{T+|T|V}{2}$. In this paper, we discuss several properties related to the spectrum, the kernel, the image, the polar decomposition of mean transform. Moreover, we investigate the image and preimage by the mean transform of some class of operators as positive, normal, unitary, hyponormal and co-hyponormal operators.

math.FA↗

arXiv2022

The Aluthge and the mean transforms of $m$-isometries

Let $T\in B(H)$ be a bounded linear operator on a Hilbert space $H$, let $T = V|T|$ be its polar decomposition of $T$ and let $λ\in [0,1]$. The $λ$-Aluthge transform $Δ_λ(T)$ and the mean transforms $M(T)$ are defined respectively by: \[Δ_λ(T):=|T|^λV|T|^{1-λ} \;\; \text{and} \;\; M(T):=\frac12(|T|V+V|T|).\] In this paper, we use several examples of weighted shift operators to prove that the Aluthge and mean transforms do not preserve the class of $m-$isometries in any directions.

math.FA↗

arXiv2022

Commuting maps with the Mean Transform under Jordan product

In this article, we give a complete characterization of the bijective maps which commute with the mean transform under Jordan product. The main result is the following : Let $H,K$ be two complex Hilbert spaces and $Φ:B(H) \to B(K)$ be a bijective map, then $$ \mathcal {M}(Φ(A)\circΦ(B))=Φ(\mathcal{M}(A\circ B)) \;\; \text{for all}\;\; A, B \in B(H)$$ if and only if there exists a unitary or anti-unitary operator $U:H\to K $ such that, $$ Φ(T)= UTU^* \; \text{for all} \;T\in B(H).$$

math.FA↗

arXiv2016

Product commuting maps with the $λ$-Aluthge transform

Let H and K be two Hilbert spaces and B(H) be the algebra of all bounded linear operators from H into itself. The main purpose of this paper is to obtain a characterization of bijective maps $Φ$ : B(H) $\rightarrow$ B(K) satisfying the following condition $Δ$ $λ$ ($Φ$(A)$Φ$(B)) = $Φ$($Δ$ $λ$ (AB)) f orall A, B $\in$ B(H), where $Δ$ $λ$ (T) stands the $λ$-Aluthge transform of the operator T $\in$ B(H). More precisely, we prove that a bijective map $Φ$ satisfies the above condition, if and only , if $Φ$(A) = U AU * for all A $\in$ B(H), for some unitary operator U : H $\rightarrow$ K.

math.FA↗

arXiv2016

New formulas for the spectral radius via Aluthge transform

In this paper we give several expressions of spectral radius of a bounded operator on a Hilbert space, in terms of iterates of Aluthge transformation, numerical radius and the asymptotic behavior of the powers of this operator. Also we obtain several characterizations of normaloid operators.

math.FA↗