• Title/Summary/Keyword: SVEP

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LOCAL SPECTRAL THEORY AND QUASINILPOTENT OPERATORS

  • YOO, JONG-KWANG
    • 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 XA(F) = XA+R(F) for all subset F ⊆ ℂ and 𝜎loc(A) = 𝜎loc(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.

GENERALIZED BROWDER, WEYL SPECTRA AND THE POLAROID PROPERTY UNDER COMPACT PERTURBATIONS

  • Duggal, Bhaggy P.;Kim, In Hyoun
    • Journal of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.281-302
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    • 2017
  • For a Banach space operator $A{\in}B(\mathcal{X})$, let ${\sigma}(A)$, ${\sigma}_a(A)$, ${\sigma}_w(A)$ and ${\sigma}_{aw}(A)$ denote, respectively, its spectrum, approximate point spectrum, Weyl spectrum and approximate Weyl spectrum. The operator A is polaroid (resp., left polaroid), if the points $iso{\sigma}(A)$ (resp., $iso{\sigma}_a(A)$) are poles (resp., left poles) of the resolvent of A. Perturbation by compact operators preserves neither SVEP, the single-valued extension property, nor the polaroid or left polaroid properties. Given an $A{\in}B(\mathcal{X})$, we prove that a sufficient condition for: (i) A+K to have SVEP on the complement of ${\sigma}_w(A)$ (resp., ${\sigma}_{aw}(A)$) for every compact operator $K{\in}B(\mathcal{X})$ is that ${\sigma}_w(A)$ (resp., ${\sigma}_{aw}(A)$) has no holes; (ii) A + K to be polaroid (resp., left polaroid) for every compact operator $K{\in}B(\mathcal{X})$ is that iso${\sigma}_w(A)$ = ∅ (resp., $iso{\sigma}_{aw}(A)$ = ∅). It is seen that these conditions are also necessary in the case in which the Banach space $\mathcal{X}$ is a Hilbert space.

Quantitative Real-Time RT-PCR of ITGA7, SVEP1, TNS1, LPHN3, SEMA3G, KLB and MMP13 mRNA Expression in Breast Cancer

  • Kotepui, Manas;Thawornkuno, Charin;Chavalitshewinkoon-Petmitr, Porntip;Punyarit, Phaibul;Petmitr, Songsak
    • Asian Pacific Journal of Cancer Prevention
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    • v.13 no.11
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    • pp.5879-5882
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    • 2012
  • Breast cancer is the leading cause of cancer deaths among women worldwide, including Thailand. In the present study, the differential mRNA expression of SVEP1, LPHN3, KLB, ITGA7, SEMA3G, TNS1 and MMP13 genes was examined in breast cancer using quantitative real-time reverse transcription polymerase chain reaction (QRT-PCR). Among these genes, increased LPHN3 and MMP13 mRNA expression levels correlated with axillary-node metastasis (P=0.02). Multiple logistic regression analysis revealed that LPHN3 and MMP13 mRNA expression is significantly associated with axillary node status in breast cancer (P=0.04).

UPPER TRIANGULAR OPERATORS WITH SVEP

  • Duggal, Bhagwati Prashad
    • Journal of the Korean Mathematical Society
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    • v.47 no.2
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    • pp.235-246
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    • 2010
  • A Banach space operator A $\in$ B(X) is polaroid if the isolated points of the spectrum of A are poles of the resolvent of A; A is hereditarily polaroid, A $\in$ ($\mathcal{H}\mathcal{P}$), if every part of A is polaroid. Let $X^n\;=\;\oplus^n_{t=i}X_i$, where $X_i$ are Banach spaces, and let A denote the class of upper triangular operators A = $(A_{ij})_{1{\leq}i,j{\leq}n$, $A_{ij}\;{\in}\;B(X_j,X_i)$ and $A_{ij}$ = 0 for i > j. We prove that operators A $\in$ A such that $A_{ii}$ for all $1{\leq}i{\leq}n$, and $A^*$ have the single-valued extension property have spectral properties remarkably close to those of Jordan operators of order n and n-normal operators. Operators A $\in$ A such that $A_{ii}$ $\in$ ($\mathcal{H}\mathcal{P}$) for all $1{\leq}i{\leq}n$ are polaroid and have SVEP; hence they satisfy Weyl's theorem. Furthermore, A+R satisfies Browder's theorem for all upper triangular operators R, such that $\oplus^n_{i=1}R_{ii}$ is a Riesz operator, which commutes with A.

ON LOCAL SPECTRAL PROPERTIES OF RIESZ OPERATORS

  • JONG-KWANG YOO
    • 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 XS(F) ∈ Lat(S), where F = {0} or F ⊆ ℂ ⧵ {0} is closed then T|XS(F) and T|XT(F) + S|XS(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, 𝜎loc, 𝜎sur, 𝜎ap}. 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.

LOCAL SPECTRAL THEORY

  • YOO, JONG-KWANG
    • 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.

LOCAL SPECTRAL PROPERTIES OF QUASI-DECOMPOSABLE OPERATORS

  • Yoo, Jong-Kwang;Oh, Heung Joon
    • Journal of the Chungcheong Mathematical Society
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    • v.29 no.4
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    • pp.543-552
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    • 2016
  • In this paper we investigate the local spectral properties of quasidecomposable operators. We show that if $T{\in}L(X)$ is quasi-decomposable, then T has the weak-SDP and ${\sigma}_{loc}(T)={\sigma}(T)$. Also, we show that the quasi-decomposability is preserved under commuting quasi-nilpotent perturbations. Moreover, we show that if $f:U{\rightarrow}{\mathbb{C}}$ is an analytic and injective on an open neighborhood U of ${\sigma}(T)$, then $T{\in}L(X)$ is quasi-decomposable if and only if f(T) is quasi-decomposable. Finally, if $T{\in}L(X)$ and $S{\in}L(Y)$ are asymptotically similar, then T is quasi-decomposable if and only if S does.

COMMON LOCAL SPECTRAL PROPERTIES OF INTERTWINING LINEAR OPERATORS

  • Yoo, Jong-Kwang;Han, Hyuk
    • 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.

BISHOP'S PROPERTY (${\beta}$) AND SPECTRAL INCLUSIONS ON BANACH SPACES

  • Yoo, Jong-Kwang;Oh, Heung-Joon
    • 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.