• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.785-794
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• 2022

In this paper we show that if A ∈ L(X) and R ∈ L(X) is a quasinilpotent operator commuting with A then X_A(F) = X_A+R(F) for all subset F ⊆ ℂ and 𝜎_loc(A) = 𝜎_loc(A + R). Moreover, we show that A and A + R share many common local spectral properties such as SVEP, property (C), property (𝛿), property (𝛽) and decomposability. Finally, we show that quasisimility preserves local spectrum.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.137-145
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• 2009

Let T ${\in}$ $\mathcal{L}$(X), S ${\in}$ $\mathcal{L}$(Y ), A ${\in}$ $\mathcal{L}$(X, Y ) and B ${\in}$ $\mathcal{L}$(Y,X) such that SA = AT, TB = BS, AB = S and BA = T. Then S and T shares that same local spectral properties SVEP, property (${\beta}$), property $({\beta})_{\epsilon}$, property (${\delta}$) and decomposability. From these common local spectral properties, we give some results related with Aluthge transforms and subscalar operators.

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.29-33
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• 1996

In this paper we shall generalize a Clary theorem by using the local spectral theory; If $ T \in L(H)$ has property $(\beta)$ and A is any operator such that $A \prec T$, then $\sigma(T) \subseteq \sigma(A)$.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.459-468
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• 2011

Let T ${\in}$ L(X), S ${\in}$ L(Y), A ${\in}$ L(X, Y) and B ${\in}$ L(Y, X) such that SA = AT, TB = BS, AB = S and BA = T. Then S and T shares the same local spectral properties SVEP, Bishop's property (${\beta}$), property $({\beta})_{\epsilon}$, property (${\delta}$) and and subscalarity. Moreover, the operators ${\lambda}I$ - T and ${\lambda}I$ - S have many basic operator properties in common.

• Journal of applied mathematics & informatics
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• v.38 no.3_4
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• pp.261-269
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• 2020

For any Banach spaces X and Y, let L(X, Y) denote the set of all bounded linear operators from X to Y. Let A ∈ L(X, Y) and B, C ∈ L(Y, X) satisfying operator equation ABA = ACA. In this paper, we prove that AC and BA share the local spectral properties such as a finite ascent, a finite descent, property (K), localizable spectrum and invariant subspace.

• Journal of applied mathematics & informatics
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• v.41 no.2
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• pp.273-286
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• 2023

In this paper we show that if T ∈ L(X) and S ∈ L(X) is a Riesz operator commuting with T and X_S(F) ∈ Lat(S), where F = {0} or F ⊆ ℂ ⧵ {0} is closed then T|X_S(F) and T|X_T(F) + S|X_S(F) share the local spectral properties such as SVEP, Dunford's property (C), Bishop's property (𝛽), decomopsition property (𝛿) and decomposability. As a corollary, if T ∈ L(X) and Q ∈ L(X) is a quasinilpotent operator commuting with T then T is Riesz if and only if T + Q is Riesz. We also study some spectral properties of Riesz operators acting on Banach spaces. We show that if T, S ∈ L(X) such that TS = ST, and Y ∈ Lat(S) is a hyperinvarinat subspace of X for which 𝜎(S|Y ) = {0} then 𝜎_*(T|Y + S|Y ) = 𝜎_*(T|Y ) for 𝜎_* ∈ {𝜎, 𝜎_loc, 𝜎_sur, 𝜎_ap}. Finally, we show that if T ∈ L(X) and S ∈ L(Y ) on the Banach spaces X and Y and T is similar to S then T is Riesz if and only if S is Riesz.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.499-507
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• 2010

In this note, we study the local spectral properties of semi-shifts. If $T\;{\in}\;L(X)$ is a semi-shift on a complex Banach space X, then T is admissible. We also prove that if $T\;{\in}\;L(X)$ is subadmissible, then $X_T(F)\;=\;E_T(F)$ for all closed $F\;{\subseteq}\;\mathbb{C}$. In particular, every subscalar operator on a Banach space is admissible.

• Communications of the Korean Mathematical Society
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• v.12 no.4
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• pp.911-920
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• 1997

This note centers around the class M(A) of multipliers on a Gelfand algebra A. This class is a large subalgebra of the Banach algebra L(A). The aim of this note is to investigate some aspects concerning their local spectral properties of multipliers. In the last part of work we consider some applications to automatic continuity theory.

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.699-710
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• 2021

In this paper, we study various connections of local spectral properties, invariant subspaces, and spectra when an operator S in 𝓛(𝓗) is an ampliation quasiaffine transform of an operator T in 𝓛(𝓗).

• Journal of the Chungcheong Mathematical Society
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• v.28 no.3
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• pp.419-429
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• 2015

In this paper, we study properties SVEP, Bishop's property (${\beta}$), property (${\delta}$), decomposability, property $({\beta})_{\epsilon}$, subscalarity, Kato spectrum, generalized Kato decomposition, finite ascent for bounded linear operators in Banach spaces.