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Snežana Č. Živković-Zlatanović

Publications and source records attributed to Snežana Č. Živković-Zlatanović.

8 recordsLinked to original sources
arXiv2024

New properties of some classes of Saphar type operators

This paper explores additional properties of some classes of Saphar type operators, namely left Drazin invertible, essentially left Drazin invertible, right Drazin invertible, and essentially right Drazin invertible operators on Banach spaces, building upon the groundwork laid in \cite{GM} and \cite{ZS}. Specifically, we propose alternative definitions for these operators and characterize them with a new type of operator decomposition, providing a deeper understanding of their properties. Furthermore, we investigate their behavior under powers. The operators we study are distinguished from other operators bearing the same name in existing literature, see \cite{Aiena}, \cite{Q. Jiang}. By employing more refined definitions, we uncover a broader range of properties for these operators, setting them apart from their counterparts in the literature.

math.FA↗
arXiv2019

On Koliha-Drazin invertible operators and Browder type theorems

Let $T$ be a bounded linear operator on a Banach space $X$. We give new necessary and sufficient conditions for $T$ to be Drazin or Koliha-Drazin invertible. All those conditions have the following form: $T$ possesses certain decomposition property and zero is not an interior point of some part of the spectrum of $T$. In addition, we study operators $T$ satisfying Browder\textquoteright s theorem, or a-Browder\textquoteright s theorem, by means of some relationships between diferent parts of the spectrum of $T$.

math.FA↗
arXiv2019

Generalized Drazin-meromorphic invertible operators and generalized Kato-meromorphic decomposition

A bounded linear operator $T$ on a Banach space $X$ is said to be generalized Drazin-meromorphic invertible if there exists a bounded linear operator $S$ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is meromorphic. We shall say that $T$ admits a generalized Kato-meromorphic decomposition if there exists a pair of $T$-invariant closed subspaces $(M,N)$ such that $X=M\oplus N$, the reduction $T_M$ is Kato and the reduction $T_N$ is meromorphic. In this paper we shall investigate such kind of operators and corresponding spectra, the generalized Drazin-meromorphic spectrum and the generalized Kato-meromorphic spectrum, and prove that these spectra are empty if and only if the operator $T$ is polynomially meromorphic. Also we obtain that the generalized Kato-meromorphic spectrum differs from the Kato type spectrum on at most countably many points. Among others, bounded linear operators which can be expressed as a direct sum of a meromorphic operator and a bounded below (resp. surjective, upper (lower) semi-Fredholm, Fredholm, upper (lower) semi-Weyl, Weyl) operator are studied. In particular, we shall characterize the single-valued extension property at a point $λ_0\in\mathbb{C}$ in the case that $λ_0-T$ admits a generalized Kato-meromorphic decomposition. As a consequence we get several results on cluster points of some distinguished parts of the spectrum.

math.SP↗
arXiv2018

Toplogical uniform descent, quasi-Fredholmness and operators originated from semi-B-Fredholm theory

In this paper we study operators originated from semi-B-Fredholm theory and as a consequence we get some results regarding boundaries and connected hulls of the corresponding spectra. In particular, we prove that a bounded linear operator $T$ acting on a Banach space, having topological uniform descent, is a {\bf BR} operator if and only if $0$ is not an accumulation point of the associated spectrum $σ_{\bf R}(T)=\{λ\in\CC:T-λI\notin {\bf R}\}$, where ${\bf R}$ denote any of the following classes: upper semi-Weyl operators, Weyl operators, upper semi-Fredholm operators, Fredholm operators, operators with finite (essential) descent and ${\bf BR}$ the B-regularity associated to ${\bf R}$ as in \cite{P8}. Under the stronger hypothesis of quasi-Fredholmness of $T,$ we obtain a similar characterization for $T$ being a {$\bf BR$} operator for much larger families of sets ${\bf R}.$

math.SP↗
arXiv2018

Pseudo-B-Fredholm operators, poles of the resolvent and mean convergence in the Calkin Algebra

We define here a pseudo B-Fredholm operator as an operator such that 0 is isolated in its essential spectrum, then we prove that an operator $T$ is pseudo- B-Fredholm if and only if $T = R + F$ where $R$ is a Riesz operator and $F$ is a B-Fredholm operator such that the commutator $[R,\, F]$ is compact. Moreover, we prove that 0 is a pole of the resolvent of an operator $T$ in the Calkin algebra if and only if $T= K+F$, where $K$ is a power compact operator and $F$ is a B-Fredholm operator, such that the commutator $[K,\, F]$ is compact. As an application, we characterize the mean convergence in the Calkin algebra.

math.SP↗
arXiv2016

Generalized Kato-Riesz decomposition

We shall say that a bounded linear operator $T$ acting on a Banach space $X$ admits a generalized Kato-Riesz decomposition if there exists a pair of $T$-invariant closed subspaces $(M,N)$ such that $X=M\oplus N$, the reduction $T_M$ is Kato and $T_N$ is Riesz. In this paper we define and investigate the generalized Kato-Riesz spectrum of an operator. For $T$ is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator $S $ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is Riesz. We investigate generalized Drazin-Riesz invertible operators and also, characterize bounded linear operators which can be expressed as a direct sum of a Riesz operator and a bounded below (resp. surjective, upper (lower) semi-Fredholm, Fredholm, upper (lower) semi-Weyl, Weyl) operator. In particular we characterize the single-valued extension property at a point $λ_0\in{\mathbb C}$ in the case that $λ_0-T$ admits a generalized Kato-Riesz decomposition.

math.FA↗
arXiv2016

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-isolated boundary point of the spectrum $σ_{\bf R}(T)$ belongs to the generalized Kato spectrum of $T$.

math.FA↗