• Title, Summary, Keyword: decomposable operator

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SPECTRA OF ASYMPTOTICALLY QUASISIMILAR SUBDECOMPOSABLE OPERATORS

  • Yoo, Jong-Kwang;Han, Hyuk
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.2
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    • pp.271-279
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    • 2009
  • In this paper, we prove that asymptotically quasisimilar sub-decomposable operators have equal spectra and quasisimilar decomposable operators have equal spectra. Moreover, every subscalar operator is admissible.

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CONTINUITY OF LINEAR OPERATOR INTERTWINING WITH DECOMPOSABLE OPERATORS AND PURE HYPONORMAL OPERATORS

  • Park, Sung-Wook;Han, Hyuk;Park, Se Won
    • Journal of the Chungcheong Mathematical Society
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    • v.16 no.1
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    • pp.37-48
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    • 2003
  • In this paper, we show that for a pure hyponormal operator the analytic spectral subspace and the algebraic spectral subspace are coincide. Using this result, we have the following result: Let T be a decomposable operator on a Banach space X and let S be a pure hyponormal operator on a Hilbert space H. Then every linear operator ${\theta}:X{\rightarrow}H$ with $S{\theta}={\theta}T$ is automatically continuous.

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WEAKLY WELL-DECOMPOSABLE OPERATORS AND AUTOMATIC CONTINUITY

  • Cho, Tae-Geun;Han, Hyuk
    • Journal of the Korean Mathematical Society
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    • v.33 no.2
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    • pp.347-365
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    • 1996
  • Let X and Y be Banach spaces and consider a linear operator $\theta : X \to Y$. The basic automatic continuity problem is to derive the continuity of $\theta$ from some prescribed algebraic conditions. For example, if $\theta : X \to Y$ is a linear operator intertwining with $T \in L(X)$ and $S \in L(Y)$, one may look for algebraic conditions on T and S which force $\theta$ to be continuous.

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PROPERTIES OF OPERATOR MATRICES

  • An, Il Ju;Ko, Eungil;Lee, Ji Eun
    • Journal of the Korean Mathematical Society
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    • v.57 no.4
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    • pp.893-913
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    • 2020
  • Let �� be the collection of the operator matrices $\(\array{A&C\\Z&B}\)$ where the range of C is closed. In this paper, we study the properties of operator matrices in the class ��. We first explore various local spectral relations, that is, the property (β), decomposable, and the property (C) between the operator matrices in the class �� and their component operators. Moreover, we investigate Weyl and Browder type spectra of operator matrices in the class ��, and as some applications, we provide the conditions for such operator matrices to satisfy a-Weyl's theorem and a-Browder's theorem, respectively.

ALGEBRAIC SPECTRAL SUBSPACES OF GENERALIZED SCALAR OPERATORS

  • Han, Hyuk
    • Communications of the Korean Mathematical Society
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    • v.9 no.3
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    • pp.617-627
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    • 1994
  • Algebraic spectral subspaces and admissible operators were introduced by K. B. Laursen and M. M. Neumann in 1988 [L88], [N]. These concepts are useful in automatic continuity problems of intertwining linear operators on Banach spaces. In this paper we characterize the algebraic spectral subspaces of generalized scalar operators. From this characterization we show that generalized scalar operators are admissible. Also we show that doubly power bounded operators are generalized scalar. And using the spectral capacity we show that a generalized scalar operator is decomposable. Then we give an example of an operator which is not admissible but decomposable.

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SOME INVARIANT SUBSPACES FOR SUBSCALAR OPERATORS

  • Yoo, Jong-Kwang
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.6
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    • pp.1129-1135
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    • 2011
  • In this note, we prove that every subscalar operator with finite spectrum is algebraic. In particular, a quasi-nilpotent subscala operator is nilpotent. We also prove that every subscalar operator with property (${\delta}$) on a Banach space of dimension greater than 1 has a nontrivial invariant closed linear subspace.

ON THE SPECTRAL MAXIMAL SPACES OF A MULTIPLICATION OPERATOR

  • Park, Jae-Chul;Yoo, Jong-Kwang
    • Journal of the Korean Mathematical Society
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    • v.33 no.1
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    • pp.205-216
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    • 1996
  • In [13], Ptak and Vrbova proved that if T is a bounded normal operator T on a complex Hilbert space H, then the ranges of the spectral projections can be represented in the form $$ \varepsilon(F)H = \bigcap_{\lambda\notinF} (T - \lambda I) H for all closed subsets F of C, $$ where $\varepsilon$ denotes the spectral measure associated with T.

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ON SPECTRAL SUBSPACES OF SEMI-SHIFTS

  • Han, Hyuk;Yoo, Jong-Kwang
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.2
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    • pp.247-257
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    • 2008
  • In this paper, we show that for a semi-shift the analytic spectral subspace coincides with the algebraic spectral subspace. Using this result, we have the following result. Let T be a decomposable operator on a Banach space ${\mathcal{X}}$ and let S be a semi-shift on a Banach space ${\mathcal{Y}}$. Then every linear operator ${\theta}:{\mathcal{X}}{\rightarrow}{\mathcal{Y}}$ with $S{\theta}={\theta}T$ is necessarily continuous.

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SOME REMARKS ON THE HELTON CLASS OF AN OPERATOR

  • Kim, In-Sook;Kim, Yoen-Ha;Ko, Eun-Gil;Lee, Ji-Eun
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.3
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    • pp.535-543
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    • 2009
  • In this paper we study some properties of the Helton class of an operator. In particular, we show that the Helton class preserves the quasinilpotent property and Dunford's boundedness condition (B). As corollaries, we get that the Helton class of some quadratically hyponormal operators or decomposable subnormal operators satisfies Dunford's boundedness condition (B).