We study the relationships between the spectra derived from Fredholm theory corresponding to two given bounded linear operators acting on the same space. The main goal of this paper is to obtain sufficient conditions for which the spectra derived from Fredholm theory and other parts of the spectra corresponding to two given operators are preserved. As an application of our results, we give conditions for which the above mentioned spectra corresponding to two multiplication operators acting on the...

Let $T$ be a Banach space operator. In this paper we characterize $a$-Browder’s theorem for $T$ by the localized single valued extension property. Also, we characterize $a$-Weyl’s theorem under the condition $E^a\left(T\right)={\pi }^a\left(T\right),$ where $E^a\left(T\right)$ is the set of all eigenvalues of $T$ which are isolated in the approximate point spectrum and ${\pi }^a\left(T\right)$ is the set of all left poles of $T.$ Some applications are also given.

Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function ${\phi }_a\left(t\right):=\phi \left({\alpha }_{t}a\right)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum ${\sigma }_w*\left({\phi }_a\right)$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define ${Ʌ}_{\phi }^a$ to be the union of all...

We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties $\left(\mathrm{SBaw}\right)$, $\left(\mathrm{SBab}\right)$, $\left(\mathrm{SBw}\right)$ and $\left(\mathrm{SBb}\right)$ are not preserved under direct sums of operators. However, we prove that if $S$ and $T$ are bounded linear operators acting on Banach spaces and having the property $\left(\mathrm{SBab}\right)$, then $S\oplus T$ has the property $\left(\mathrm{SBab}\right)$ if and only if ${\sigma }_{{\mathrm{SBF}}_{+}^{-}}(S\oplus T)={\sigma }_{{\mathrm{SBF}}_{+}^{-}}\left(S\right)\cup {\sigma }_{{\mathrm{SBF}}_{+}^{-}}\left(T\right)$, where ${\sigma }_{{\mathrm{SBF}}_{+}^{-}}\left(T\right)$ is the upper semi-B-Weyl spectrum of $T$. We obtain analogous preservation results for the properties $\left(\mathrm{SBaw}\right)$, $\left(\mathrm{SBb}\right)$ and $\left(\mathrm{SBw}\right)$ with...

In this paper, we give a new approach to the study of Weyl-type theorems. Precisely, we introduce the concepts of spectral valued and spectral partitioning functions. Using two natural order relations on the set of spectral valued functions, we reduce the question of relationship between Weyl-type theorems to the study of the set difference between the parts of the spectrum that are involved. This study solves completely the question of relationship between two spectral valued functions, comparable...

We study the essential ascent and the related essential ascent spectrum of an operator on a Banach space. We show that a Banach space X has finite dimension if and only if the essential ascent of every operator on X is finite. We also focus on the stability of the essential ascent spectrum under perturbations, and we prove that an operator F on X has some finite rank power if and only if ${\sigma }_{asc}^e(T+F)={\sigma }_{asc}^e\left(T\right)$ for every operator T commuting with F. The quasi-nilpotent part, the analytic core and the single-valued extension...