Snezana Zivkovic-Zlatanovic - Academia.edu (original) (raw)

Papers by Snezana Zivkovic-Zlatanovic

Research paper thumbnail of Generalized Kato-Riesz decomposition

arXiv: Functional Analysis, 2016

We shall say that a bounded linear operator TTT acting on a Banach space XXX admits a generalized... more We shall say that a bounded linear operator TTT acting on a Banach space XXX admits a generalized Kato-Riesz decomposition if there exists a pair of TTT-invariant closed subspaces (M,N)(M,N)(M,N) such that X=MoplusNX=M\oplus NX=MoplusN, the reduction TMT_MTM is Kato and TNT_NTN is Riesz. In this paper we define and investigate the generalized Kato-Riesz spectrum of an operator. For TTT is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator SS S acting on XXX such that TS=STTS=STTS=ST, STS=SSTS=SSTS=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 lambda_0inmathbbC\lambda_0\in{\mathbb C}lambda0inmathbbC in the case that lambda0−T\lambda_0-Tlambda_0−T admits a generalized Kato-Riesz decomposition.

Research paper thumbnail of Drazin Invertibility Relative to Some Subsets of Quasinilpotents and Homomorphism Ranges

Results in Mathematics, Feb 18, 2023

Research paper thumbnail of Perturbations, Quasinilpotent Equivalence and Communicating Operators

Mathematical proceedings of the Royal Irish Academy, 2015

Research paper thumbnail of Generalized Drazin invertible elements relative to a regularity

Linear & Multilinear Algebra, Feb 23, 2023

Research paper thumbnail of Generalized Drazin-meromorphic invertible operators and generalized Kato-meromorphic decomposition

arXiv (Cornell University), Apr 9, 2019

A bounded linear operator T on a Banach space X is said to be generalized Drazinmeromorphic inver... more A bounded linear operator T on a Banach space X is said to be generalized Drazinmeromorphic invertible if there exists a bounded linear operator S acting on X such that T S = ST , ST S = S, T ST − 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 ⊕ N , the reduction TM is Kato and the reduction TN 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 Katomeromorphic 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 λ0 ∈ C in the case that λ0 − T admits a generalized Kato-meromorphic decomposition. As a consequence we get several results on cluster points of some distinguished parts of the spectrum.

Research paper thumbnail of On Koliha-Drazin invertible operators and Browder type theorems

arXiv (Cornell University), Dec 1, 2019

Let T be a bounded linear operator on a Banach space X. We give new necessary and sufficient cond... more 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's theorem, or a-Browder's theorem, by means of some relationships between diferent parts of the spectrum of T .

Research paper thumbnail of Toplogical uniform descent, quasi-Fredholmness and operators originated from semi-B-Fredholm theory

arXiv (Cornell University), Jun 9, 2018

In this paper we study operators originated from semi-B-Fredholm theory and as a consequence we g... more 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 BR operator if and only if 0 is not an accumulation point of the associated spectrum σR(T) = {λ ∈ C : T − λI / ∈ R}, where 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 BR the B-regularity associated to R as in [6]. Under the stronger hypothesis of quasi-Fredholmness of T, we obtain a similar characterisation for T being a BR operator for much larger families of sets R.

Research paper thumbnail of Generalized Kato-meromorphic decomposition, generalized Drazin-meromorphic invertible operators and single-valued extension property

Banach Journal of Mathematical Analysis, 2020

In this paper we define and study generalized Kato-meromorphic decomposition and generalized Draz... more In this paper we define and study generalized Kato-meromorphic decomposition and generalized Drazin-meromorphic invertible operators. 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. Among others, we show that T is generalized Drazin-meromorphic invertible if and only if T admits a generalized Kato-meromorphic decomposition and 0 is not an interior point of \(\sigma (T)\), and this is also equivalent to the fact that T is a direct sum of a meromorphic operator and an invertible operator. Also we study 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. In particular, we characterize the single-valued extension property at a point \(\lambda _0\in \mathbb {C}\) in the case that \(\lambda _0-T\) admits a generalized Kato-meromorphic decomposition, and as a consequence we get several results on cluster points of some distinguished part of the spectrum. Furthermore, we investigate corresponding spectra and prove that these spectra are empty if and only if the operator T is polynomially meromorphic.

Research paper thumbnail of On some classes of Saphar type operators

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

Research paper thumbnail of Ruston, Riesz and perturbation classes

Journal of Mathematical Analysis and Applications, 2012

We determine the perturbation classes of Fredholm and Weyl elements, as well the "commuting pertu... more We determine the perturbation classes of Fredholm and Weyl elements, as well the "commuting perturbation classes" of Fredholm, Weyl and Browder elements with respect to unbounded Banach algebra homomorphism T. Among other things we use Ruston elements of Mouton, Mouton and Raubenheimer. Also, we investigate the class of polynomially almost T null and the class of polynomially T Riesz elements. 2010 Mathematics subject classification: 46H05, 47A53, 47A55.

Research paper thumbnail of A Note on Essential Spectra of Linear Operator Pencils

Bulletin of the Iranian Mathematical Society

Research paper thumbnail of Generalized Kato decomposition, Kato type decomposition and various types of spectra

arXiv: Functional Analysis, 2016

Let bfR{\bf R}bfR denote any of the following classes: invertible operators, bounded below operators,... more Let bfR{\bf R}bfR denote any of the following classes: invertible operators, bounded below operators, surjective operators, upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Browder operators, Browder operators, upper (lower) semi-Weyl operators, Weyl operators. For a bounded linear operator TTT on a Banach space XXX we show that T=TMoplusTNT=T_M\oplus T_NT=TMoplusTN with TMinbfRT_M \in {\bf R}TMinbfR and TNT_NTN quasinilpotent (nilpotent) if and only if TTT admits a generalized Kato decomposition ($T$ is of Kato type) and 000 is not an interior point of the corresponding spectrum sigmabfR(T)=lambdainmathbbC:T−lambdanotinbfR\sigma_{\bf R}(T)=\{\lambda \in \mathbb{C}: T-\lambda \notin {\bf R}\}sigmabfR(T)=lambdainmathbbC:T−lambdanotinbfR. As an application we obtain that every non-isolated boundary point of the spectrum sigmabfR(T)\sigma_{\bf R}(T)sigmabfR(T) belongs to the generalized Kato spectrum of TTT. In addition, we prove that the boundary of the generalized Drazin spectrum of TTT is contained in the generalized Kato spectrum of TTT and also, the boundary of the Drazin spectrum of TTT is co...

Research paper thumbnail of A Note on Koliha-Drazin Invertible Operators and a-Browder’s Theorem

Complex Analysis and Operator Theory, 2021

Let T be a bounded linear operator on a Banach space X. We prove certain inclusions and equalitie... more Let T be a bounded linear operator on a Banach space X. We prove certain inclusions and equalities between different parts of the spectrum of T and then apply them to study Koliha-Drazin invertible operators and operators satisfying a-Browder’s theorem.

Research paper thumbnail of Topological Uniform Descent, Quasi-Fredholmness and Operators Originated from Semi-B-Fredholm Theory

Complex Analysis and Operator Theory, 2019

In this paper we study operators originated from semi-B-Fredholm theory and as a consequence we g... more 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 BR operator if and only if 0 is not an accumulation point of the associated spectrum σR(T) = {λ ∈ C : T − λI / ∈ R}, where 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 BR the B-regularity associated to R as in [6]. Under the stronger hypothesis of quasi-Fredholmness of T, we obtain a similar characterisation for T being a BR operator for much larger families of sets R.

Research paper thumbnail of Generalized Kato Decomposition and Essential Spectra

Complex Analysis and Operator Theory, 2016

Let R denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operat... more Let 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 ⊕ T N with T M ∈ 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 R (T) = {λ ∈ C : T − λ / ∈ R}. In addition, we show that every non-isolated boundary point of the spectrum σ R (T) belongs to the generalized Kato spectrum of T .

Research paper thumbnail of Polynomially Meromorphic Operators

Mathematical Proceedings of the Royal Irish Academy, 2016

Research paper thumbnail of Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators

Linear and Multilinear Algebra, 2016

We shall say that a bounded linear operator T acting on a Banach space X admits a generalized Kat... more 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 ⊕ 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 ∈ C in the case that λ 0 − T admits a generalized Kato-Riesz decomposition.

Research paper thumbnail of On simple permanence

Quaestiones Mathematicae, 2015

Simple permanence" is one of several variants of "spectral permanence", which are curiously inter... more Simple permanence" is one of several variants of "spectral permanence", which are curiously interrelated. 2010 Mathematics Subject Classification. 46H05, 47A05, 47A53. Keywords and Phrases. Simply polar elements, semigroup homomorphisms. This research is supported by the Ministry of Science and Technological Development of Serbia, grant no. 174007 0. Introduction This is a reworking of our previous note [DZH], in which we deployed "Drazin permanence" and quasipolar Banach algebra elements in the proof of a variant of the "spectral permanence" enjoyed by C* algebras. Here we use instead "simple permanence" and simply polar elements of semigroups and rings: we believe that the argument is now more transparent and more elementary. 1. Generalized permanence If T : A → B is a "semigroup homomorphism" [DZH] then there is inclusion

Research paper thumbnail of Polynomially Riesz perturbations

Journal of Mathematical Analysis and Applications, 2013

In this paper we investigate perturbation of left (right) Fredholm, Weyl and Browder operators by... more In this paper we investigate perturbation of left (right) Fredholm, Weyl and Browder operators by polynomially Riesz operators. We show how Baklouti's idea of "communication" enhances the perturbation properties of polynomially Riesz operators.

Research paper thumbnail of Spectral permanence for the Moore-Penrose inverse

Proceedings of the American Mathematical Society, 2012

Research paper thumbnail of Generalized Kato-Riesz decomposition

arXiv: Functional Analysis, 2016

We shall say that a bounded linear operator TTT acting on a Banach space XXX admits a generalized... more We shall say that a bounded linear operator TTT acting on a Banach space XXX admits a generalized Kato-Riesz decomposition if there exists a pair of TTT-invariant closed subspaces (M,N)(M,N)(M,N) such that X=MoplusNX=M\oplus NX=MoplusN, the reduction TMT_MTM is Kato and TNT_NTN is Riesz. In this paper we define and investigate the generalized Kato-Riesz spectrum of an operator. For TTT is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator SS S acting on XXX such that TS=STTS=STTS=ST, STS=SSTS=SSTS=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 lambda_0inmathbbC\lambda_0\in{\mathbb C}lambda0inmathbbC in the case that lambda0−T\lambda_0-Tlambda_0−T admits a generalized Kato-Riesz decomposition.

Research paper thumbnail of Drazin Invertibility Relative to Some Subsets of Quasinilpotents and Homomorphism Ranges

Results in Mathematics, Feb 18, 2023

Research paper thumbnail of Perturbations, Quasinilpotent Equivalence and Communicating Operators

Mathematical proceedings of the Royal Irish Academy, 2015

Research paper thumbnail of Generalized Drazin invertible elements relative to a regularity

Linear & Multilinear Algebra, Feb 23, 2023

Research paper thumbnail of Generalized Drazin-meromorphic invertible operators and generalized Kato-meromorphic decomposition

arXiv (Cornell University), Apr 9, 2019

A bounded linear operator T on a Banach space X is said to be generalized Drazinmeromorphic inver... more A bounded linear operator T on a Banach space X is said to be generalized Drazinmeromorphic invertible if there exists a bounded linear operator S acting on X such that T S = ST , ST S = S, T ST − 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 ⊕ N , the reduction TM is Kato and the reduction TN 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 Katomeromorphic 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 λ0 ∈ C in the case that λ0 − T admits a generalized Kato-meromorphic decomposition. As a consequence we get several results on cluster points of some distinguished parts of the spectrum.

Research paper thumbnail of On Koliha-Drazin invertible operators and Browder type theorems

arXiv (Cornell University), Dec 1, 2019

Let T be a bounded linear operator on a Banach space X. We give new necessary and sufficient cond... more 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's theorem, or a-Browder's theorem, by means of some relationships between diferent parts of the spectrum of T .

Research paper thumbnail of Toplogical uniform descent, quasi-Fredholmness and operators originated from semi-B-Fredholm theory

arXiv (Cornell University), Jun 9, 2018

In this paper we study operators originated from semi-B-Fredholm theory and as a consequence we g... more 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 BR operator if and only if 0 is not an accumulation point of the associated spectrum σR(T) = {λ ∈ C : T − λI / ∈ R}, where 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 BR the B-regularity associated to R as in [6]. Under the stronger hypothesis of quasi-Fredholmness of T, we obtain a similar characterisation for T being a BR operator for much larger families of sets R.

Research paper thumbnail of Generalized Kato-meromorphic decomposition, generalized Drazin-meromorphic invertible operators and single-valued extension property

Banach Journal of Mathematical Analysis, 2020

In this paper we define and study generalized Kato-meromorphic decomposition and generalized Draz... more In this paper we define and study generalized Kato-meromorphic decomposition and generalized Drazin-meromorphic invertible operators. 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. Among others, we show that T is generalized Drazin-meromorphic invertible if and only if T admits a generalized Kato-meromorphic decomposition and 0 is not an interior point of \(\sigma (T)\), and this is also equivalent to the fact that T is a direct sum of a meromorphic operator and an invertible operator. Also we study 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. In particular, we characterize the single-valued extension property at a point \(\lambda _0\in \mathbb {C}\) in the case that \(\lambda _0-T\) admits a generalized Kato-meromorphic decomposition, and as a consequence we get several results on cluster points of some distinguished part of the spectrum. Furthermore, we investigate corresponding spectra and prove that these spectra are empty if and only if the operator T is polynomially meromorphic.

Research paper thumbnail of On some classes of Saphar type operators

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

Research paper thumbnail of Ruston, Riesz and perturbation classes

Journal of Mathematical Analysis and Applications, 2012

We determine the perturbation classes of Fredholm and Weyl elements, as well the "commuting pertu... more We determine the perturbation classes of Fredholm and Weyl elements, as well the "commuting perturbation classes" of Fredholm, Weyl and Browder elements with respect to unbounded Banach algebra homomorphism T. Among other things we use Ruston elements of Mouton, Mouton and Raubenheimer. Also, we investigate the class of polynomially almost T null and the class of polynomially T Riesz elements. 2010 Mathematics subject classification: 46H05, 47A53, 47A55.

Research paper thumbnail of A Note on Essential Spectra of Linear Operator Pencils

Bulletin of the Iranian Mathematical Society

Research paper thumbnail of Generalized Kato decomposition, Kato type decomposition and various types of spectra

arXiv: Functional Analysis, 2016

Let bfR{\bf R}bfR denote any of the following classes: invertible operators, bounded below operators,... more Let bfR{\bf R}bfR denote any of the following classes: invertible operators, bounded below operators, surjective operators, upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Browder operators, Browder operators, upper (lower) semi-Weyl operators, Weyl operators. For a bounded linear operator TTT on a Banach space XXX we show that T=TMoplusTNT=T_M\oplus T_NT=TMoplusTN with TMinbfRT_M \in {\bf R}TMinbfR and TNT_NTN quasinilpotent (nilpotent) if and only if TTT admits a generalized Kato decomposition ($T$ is of Kato type) and 000 is not an interior point of the corresponding spectrum sigmabfR(T)=lambdainmathbbC:T−lambdanotinbfR\sigma_{\bf R}(T)=\{\lambda \in \mathbb{C}: T-\lambda \notin {\bf R}\}sigmabfR(T)=lambdainmathbbC:T−lambdanotinbfR. As an application we obtain that every non-isolated boundary point of the spectrum sigmabfR(T)\sigma_{\bf R}(T)sigmabfR(T) belongs to the generalized Kato spectrum of TTT. In addition, we prove that the boundary of the generalized Drazin spectrum of TTT is contained in the generalized Kato spectrum of TTT and also, the boundary of the Drazin spectrum of TTT is co...

Research paper thumbnail of A Note on Koliha-Drazin Invertible Operators and a-Browder’s Theorem

Complex Analysis and Operator Theory, 2021

Let T be a bounded linear operator on a Banach space X. We prove certain inclusions and equalitie... more Let T be a bounded linear operator on a Banach space X. We prove certain inclusions and equalities between different parts of the spectrum of T and then apply them to study Koliha-Drazin invertible operators and operators satisfying a-Browder’s theorem.

Research paper thumbnail of Topological Uniform Descent, Quasi-Fredholmness and Operators Originated from Semi-B-Fredholm Theory

Complex Analysis and Operator Theory, 2019

In this paper we study operators originated from semi-B-Fredholm theory and as a consequence we g... more 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 BR operator if and only if 0 is not an accumulation point of the associated spectrum σR(T) = {λ ∈ C : T − λI / ∈ R}, where 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 BR the B-regularity associated to R as in [6]. Under the stronger hypothesis of quasi-Fredholmness of T, we obtain a similar characterisation for T being a BR operator for much larger families of sets R.

Research paper thumbnail of Generalized Kato Decomposition and Essential Spectra

Complex Analysis and Operator Theory, 2016

Let R denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operat... more Let 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 ⊕ T N with T M ∈ 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 R (T) = {λ ∈ C : T − λ / ∈ R}. In addition, we show that every non-isolated boundary point of the spectrum σ R (T) belongs to the generalized Kato spectrum of T .

Research paper thumbnail of Polynomially Meromorphic Operators

Mathematical Proceedings of the Royal Irish Academy, 2016

Research paper thumbnail of Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators

Linear and Multilinear Algebra, 2016

We shall say that a bounded linear operator T acting on a Banach space X admits a generalized Kat... more 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 ⊕ 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 ∈ C in the case that λ 0 − T admits a generalized Kato-Riesz decomposition.

Research paper thumbnail of On simple permanence

Quaestiones Mathematicae, 2015

Simple permanence" is one of several variants of "spectral permanence", which are curiously inter... more Simple permanence" is one of several variants of "spectral permanence", which are curiously interrelated. 2010 Mathematics Subject Classification. 46H05, 47A05, 47A53. Keywords and Phrases. Simply polar elements, semigroup homomorphisms. This research is supported by the Ministry of Science and Technological Development of Serbia, grant no. 174007 0. Introduction This is a reworking of our previous note [DZH], in which we deployed "Drazin permanence" and quasipolar Banach algebra elements in the proof of a variant of the "spectral permanence" enjoyed by C* algebras. Here we use instead "simple permanence" and simply polar elements of semigroups and rings: we believe that the argument is now more transparent and more elementary. 1. Generalized permanence If T : A → B is a "semigroup homomorphism" [DZH] then there is inclusion

Research paper thumbnail of Polynomially Riesz perturbations

Journal of Mathematical Analysis and Applications, 2013

In this paper we investigate perturbation of left (right) Fredholm, Weyl and Browder operators by... more In this paper we investigate perturbation of left (right) Fredholm, Weyl and Browder operators by polynomially Riesz operators. We show how Baklouti's idea of "communication" enhances the perturbation properties of polynomially Riesz operators.

Research paper thumbnail of Spectral permanence for the Moore-Penrose inverse

Proceedings of the American Mathematical Society, 2012