Imen FERJANI

In this paper, we introduce new pseudo-spectral sets called the left (right) γ-pseudospectrum and left (right) γ-ε-condition pseudospectrum of bounded linear operators on Banach spaces. We provide a sufficient condition for invertibility using the Kuratowski measure of noncompactness, and establish characterizations of these new spectral sets.

This work introduces two new classes of operators, called order L-weakly demicompact (order LWDC) and order M-weakly demicompact (order MWDC). We give several results that compare order LW-compact and order MW-compact operators with the corresponding demicompact operators. We also define order power MW-compact operators and prove that they belong to the class of order MWDC operators. In addition, we show that this class is preserved under perturbations by order M-weakly compact operators.

Study of relatively generalized weakly demicompact (GWMD) operators on Banach spaces. New characterizations via De Blasi measure of noncompactness, stability under perturbations, analysis of 2×2 block operator matrices, and numerical experiments on weighted-shift and Volterra-type operators.

Introduction of p-M-weakly demicompact operators (PMWD) within lattice-normed vector lattices, generalizing weak compactness and demicompactness. Relationships with p-compact and p-M-weakly compact operators, stability under perturbations, and a numerical application to hidden state stability in neural networks.

Stability of generalized weakly demicompact operators on Banach spaces under strictly singular and cosingular perturbations, via the notions of perturbation and coperturbation functions. Results applied to establish invariance of the essential spectrum.

Spectral properties of unbounded generalized Fredholm operators on non-reflexive Banach spaces. Introduction of g-g-Riesz linear operators as an extension of Riesz operators, with applications to generalized essential spectra and perturbation results.

Introduction of p-demicompact operators between lattice normed spaces, with conditions under which p-bounded operators are p-demicompact, and a study of the modulus of p-demicompact operators.

Index theory for generalized Fredholm operators extending classical Fredholm index theory. Perturbation results and index stability under compact and generalized perturbations.

Extension of generalized weak demicompactness using the measure of non-strict-singularity. New characterizations and stability theorems for essential spectra derived under this generalized compactness condition.

De Gruyter volume chapter proving invariance of essential spectrum under perturbations by generalized weakly demicompact operators, extending spectral stability to broad classes of operators in Banach spaces.

Detailed study introducing generalized weakly S-demicompact operators and deriving results on essential spectra, Fredholm theory, and perturbation in Banach space settings.

Analysis of relative essential spectra of 3×3 operator matrices, proving invariance and stability results via relative generalized weak demicompactness, combining block operator matrix methods with Banach space spectral analysis.

Foundational paper introducing generalized weakly demicompact operators. Key results include characterizations via Fredholm theory and invariance of essential spectra, establishing the basis for a research programme on operator compactness generalizations.

Under review. Develops characterizations of the γ-pseudo left-right Browder essential spectra for operators on Banach spaces, establishing stability results under Riesz perturbations and extending pseudospectral theory to generalized Browder settings.

Work in progress. Introduces and investigates order-demi quasi KB operators in Banach lattice settings, aiming to unify and extend existing operator classes with applications to spectral and lattice-theoretic analysis.