### A Characterization of Riesz Operators.

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We characterize some S-essential spectra of a closed linear relation in terms of certain linear relations of semi-Fredholm type.

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.

We investigate the Banach manifold consisting of complex ${}^{r}$ functions on the unit disc having boundary values in a given one-dimensional submanifold of the plane. We show that ∂/∂λ̅ restricted to that submanifold is a Fredholm mapping. Moreover, for any such function we obtain a relation between its homotopy class and the Fredholm index.

From Corollary 3.5 in [Berkani, M; Sarih, M.; Studia Math. 148 (2001), 251–257] we know that if $S$, $T$ are commuting $B$-Fredholm operators acting on a Banach space $X$, then $ST$ is a $B$-Fredholm operator. In this note we show that in general we do not have $error\left(ST\right)=error\left(S\right)+error\left(T\right)$, contrarily to what has been announced in Theorem 3.2 in [Berkani, M; Proc. Amer. Math. Soc. 130 (2002), 1717–1723]. However, if there exist $U,V\in L\left(X\right)$ such that $S$, $T$, $U$, $V$ are commuting and $US+VT=I$, then $error\left(ST\right)=error\left(S\right)+error\left(T\right)$, where $error$ stands for the index of a $B$-Fredholm operator.

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...

Let X be a Banach space and let T be a bounded linear operator acting on X. Atkinson's well known theorem says that T is a Fredholm operator if and only if its projection in the algebra L(X)/F₀(X) is invertible, where F₀(X) is the ideal of finite rank operators in the algebra L(X) of bounded linear operators acting on X. In the main result of this paper we establish an Atkinson-type theorem for B-Fredholm operators. More precisely we prove that T is a B-Fredholm operator if and only if its projection...

We derive a formula for the index of Fredholm chains on normed spaces.