Narrow Results

The paper aims to extend the concept of Fredholm, Weyl and Jeribi essential spectra in the quaternionic setting. Furthermore, some properties and stability of the corresponding spectra of Fredholm and Weyl operators have been investigated in this setting. To achieve the goal, a characterization of the sum of two invariant bounded linear operators has been obtained in order to explore various properties of the Fredholm operator and Weyl operator under some assumptions in quaternionic setting. Also, various sequential properties of the pseudo-resolvent operator, right quaternionic linear operator, Weyl operator, Weyl S-spectrum, Jeribi essential S-spectrum and some properties of $2\times 2$ block operator matrices have been discussed. The spectral mapping theorem of essential S-spectrum, Weyl S-spectrum and Jeribi essential S-spectrum for self-adjoint operators has been established. A characterization of the essential S-spectrum and Weyl S-spectrum of the sum of two bounded linear operators concludes this investigation.

This work is devoted to the study of semi-linear elliptic systems in unbounded cylinders with linear dependence of the components of the nonlinearity vector. We reduce the study of such a problem with non-Fredholm operator to the study of a perturbation of some reaction-diffusion operator which satisfies the Fredholm property. Then sufficient conditions that ensure the structural stability of particular solutions are given. These conditions are applied to derive some existence results for some combustion model with complex chemistry and for some KPP like system.

The paper introduces the left and right versions of the large class of Drazin inverses in terms of the left and right annihilators in a ring, which are called left-Drazin and right-Drazin inverses. We characterize some basic properties of these one-sided Drazin inverses, and discuss Jacobson’s lemma for them. In addition, the relation between the Drazin inverses and these two one-sided inverses is given by means of the spectrum and the operator decomposition. As an application, the left-Drazin and right-Drazin invertibilities in the Calkin algebra are investigated.

In connection with Fredholmness of singular integral operators in weighted generalized Hölder spaces , there is studied a certain subclass of monotonic functions ω(x) of the class of Zygmund-Bary-Stechkin which may oscillate between two power functions. In particular, there are derived explicit formulas for the best lower and upper exponents of the power-type estimation of ω(x), these exponents in general not coinciding with Boyd-type indices of ω(x).