Generalized Browder's and Weyl's Theorems for Generalized Derivations

Perturbation of Browder Spectrum of Upper-triangular Operator Matrices

Let $M_{C}=\left(\begin{array}{cc}A&C\\0&B\\\end{array} \right)$ is a 2-by-2 upper triangular operator matrix acting on the Banach space $X\oplus Y$ or Hilbert space $H\oplus K$. For the most import spectra such as spectrum, essential spectrum and Weyl spectrum, the characterizations of the perturbations of $M_{C}$ on Banach space have been presented. However, the characterization for the Browder spectrum on the Banach space is still unknown. The goal of this paper is to present some necessary and sufficient conditions for $M_C$ to be Browder for some $C\in B(Y,X)$ by using an alternative appr…

Upper Triangular Operator Matrices, SVEP and Browder. Weyl Theorems

A Banach space operator $T\in B({\cal X})$ is polaroid if points $λ\in\isoσσ(T)$ are poles of the resolvent of $T$. Let $σ_a(T)$, $σ_w(T)$, $σ_{aw}(T)$, $σ_{SF_+}(T)$ and $σ_{SF_-}(T)$ denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi--Fredholm and lower semi--Fredholm spectrum of $T$. For $A$, $B$ and $C\in B({\cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $π_0(M_C)=\{λ\in\isoσ(M_C) 0<\dim(M_C-λ)^{-1}(0)<\infty\}$, $M_0$ satisfies Weyl's theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$…

Property (z); direct sums and a note on an a-Browder type theorem

We characterize the properties $(z)$ and $(az)$ for an operator $T$ whose dual $T^*$ has the SVEP on the complementary of the upper semi-Weyl spectrum of $T.$ If $S$ and $T$ are Banach space operators satisfying property $(z)$ or $(az),$ we give conditions on $S$ and $T$ to ensure the preservation of these properties by the direct sum $S\oplus T.$ Some results are given for multipliers and in general for $(H)$-operators. Also we give a correct proof of \cite[Theorem 2.3]{SZ} which was proved by using the equality $σ_p^0(S\oplus T)= σ_p^0(S)\cup σ_p^0(T).$ However this equality is not true; we …

Generalized Kato decomposition and essential spectra

Let ${\bf R}$ denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear operator $T$ on a Banach space $X$ we prove that $T=T_M\oplus T_N$ with $T_M \in {\bf R}$ and $T_N$ quasinilpotent (nilpotent) if and only if $T$ admits a generalized Kato decomposition ($T$ is of Kato type) and $0$ is not an interior point of the corresponding spectrum $σ_{\bf R}(T)=\{λ\in \mathbb{C}: T-λ\notin {\bf R}\}$. In addition, we show that every non…

On the property $(Z_{E_a})$

The paper introduces the notion of properties $(Z_{Π_a})$ and $(Z_{E_a})$ as variants of Weyl's theorem and Browder's theorem for bounded linear operators acting on infinite dimensional Banach spaces. A characterization of these properties in terms of localized single valued extension property is given, and the perturbation by commuting Riesz operators is also studied. Classes of operators are considered as illustrating examples.

On $n$th roots of normal operators

For $n$-normal operators $A$ [2, 4, 5], equivalently $n$-th roots $A$ of normal Hilbert space operators, both $A$ and $A^*$ satisfy the Bishop--Eschmeier--Putinar property $(β)_ε$, $A$ is decomposable and the quasi-nilpotent part $H_0(A-λ)$ of $A$ satisfies $H_0(A-λ)^{-1}(0)=(A-λ)^{-1}(0)$ for every non-zero complex $λ$. $A$ satisfies every Weyl and Browder type theorem, and a sufficient condition for $A$ to be normal is that either $A$ is dominant or $A$ is a class ${\mathcal A}(1,1)$ operator.

Spectra originated from semi-B-Fredholm theory and commuting perturbations

Burgos, Kaidi, Mbekhta and Oudghiri provided an affirmative answer to a question of Kaashoek and Lay and proved that an operator $F$ is power finite rank if and only if $σ_{dsc}(T+F) =σ_{dsc}(T)$ for every operator $T$ commuting with $F$. Later, several authors extended this result to the essential descent spectrum, the left Drazin spectrum and the left essentially Drazin spectrum. In this paper, using the theory of operator with eventual topological uniform descent and the technique used in Burgos, Kaidi, Mbekhta, and Oudghiri, we generalize this result to various spectra originated from seni…

在同态李Banach代数中的谱理论与几乎周期结构

Spectral Theory and Almost Periodic Structures in Hom--Lie Banach Algebras

本文提出Hom-李Banach代数的谱理论框架，揭示了扭曲导数的谱分解及几乎周期子空间的结构特性，并通过具体例子展示了其在算子代数中的应用。