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Using the holomorphic functional calculus we give a characterization of idempotent elements commuting with a given element in a Banach algebra.

Let T be a bounded linear operator acting on a Banach space X. For each integer n, define $T_n$ to be the restriction of T to $R\left(T^n\right)$ viewed as a map from $R\left(T^n\right)$ into $R\left(T^n\right)$. In [1] and [2] we have characterized operators T such that for a given integer n, the operator $T_n$ is a Fredholm or a semi-Fredholm operator. We continue those investigations and we study the cases where $T_n$ belongs to a given regularity in the sense defined by Kordula and Müller in[10]. We also consider the regularity of operators with topological...

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...

An operator $T$ acting on a Banach space $X$ possesses property $\left(\mathrm{gw}\right)$ if ${\sigma }_a\left(T\right)\setminus {\sigma }_{{\mathrm{SBF}}_{+}^{-}}\left(T\right)=E\left(T\right),$ where ${\sigma }_a\left(T\right)$ is the approximate point spectrum of $T$, ${\sigma }_{{\mathrm{SBF}}_{+}^{-}}\left(T\right)$ is the essential semi-B-Fredholm spectrum of $T$ and $E\left(T\right)$ is the set of all isolated eigenvalues of $T.$ In this paper we introduce and study two new properties $\left(\mathrm{b}\right)$ and $\left(\mathrm{gb}\right)$ in connection with Weyl type theorems, which are analogous respectively to Browder’s theorem and generalized Browder’s theorem. Among other, we prove that if $T$ is a bounded linear operator acting on a Banach space $X$, then...

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.

Let $T$ be an operator acting on a Banach space $X$, let $\sigma \left(T\right)$ and ${\sigma }_{BW}\left(T\right)$ be respectively the spectrum and the B-Weyl spectrum of $T$. We say that $T$ satisfies the generalized Weyl’s theorem if ${\sigma }_{BW}\left(T\right)=\sigma \left(T\right)\setminus E\left(T\right)$, where $E\left(T\right)$ is the set of all isolated eigenvalues of $T$. The first goal of this paper is to show that if $T$ is an operator of topological uniform descent and $0$ is an accumulation point of the point spectrum of $T,$ then $T$ does not have the single valued extension property at $0$, extending an earlier result of J. K. Finch and a...