• Title/Summary/Keyword: M-ideal

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Operators in L(X,Y) in which K(X,Y) is a semi M-ideal

  • Cho, Chong-Man
    • Bulletin of the Korean Mathematical Society
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    • v.29 no.2
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    • pp.257-264
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    • 1992
  • Since Alfsen and Effors [1] introduced the notion of an M-ideal, many authors [3,6,9,12] have worked on the problem of finding those Banach spaces X and Y for which K(X,Y), the space of all compact linear operators from X to Y, is an M-ideal in L(X,Y), the space of all bounded linear operators from X to Y. The M-ideal property of K(X,Y) in L(X,Y) gives some informations on X,Y and K(X,Y). If K(X) (=K(X,X)) is an M-ideal in L(X) (=L(X,X)), then X has the metric compact approximation property [5] and X is an M-ideal in $X^{**}$ [10]. If X is reflexive and K(X) is an M-ideal in L(X), then K(X)$^{**}$ is isometrically isomorphic to L(X)[5]. A weaker notion is a semi M-ideal. Studies on Banach spaces X and Y for which K(X,Y) is a semi M-ideal in L(X,Y) were done by Lima [9, 10].

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On M-ideal properties of certain spaces of compact operators

  • Cho, Chong-Man;Kim, Beom-Sool
    • Communications of the Korean Mathematical Society
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    • v.11 no.3
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    • pp.673-680
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    • 1996
  • It is proved that $K(c_0,Y)$ is an M-ideal in $L(c_0,Y)$ if Y is a closed subspace of $c_0$. And a new direct proof of the fact that $K(L_1[0,1],\ell_1)$ is not an M-ideal in $L(L_1[0,1],\ell_1)$ is given.

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HEREDITARY PROPERTIES OF CERTAIN IDEALS OF COMPACT OPERATORS

  • Cho, Chong-Man;Lee, Eun-Joo
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.3
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    • pp.457-464
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    • 2004
  • Let X be a Banach space and Z a closed subspace of a Banach space Y. Denote by L(X, Y) the space of all bounded linear operators from X to Y and by K(X, Y) its subspace of compact linear operators. Using Hahn-Banach extension operators corresponding to ideal projections, we prove that if either $X^{**}$ or $Y^{*}$ has the Radon-Nikodym property and K(X, Y) is an M-ideal (resp. an HB-subspace) in L(X, Y), then K(X, Z) is also an M-ideal (resp. HB-subspace) in L(X, Z). If L(X, Y) has property SU instead of being an M-ideal in L(X, Y) in the above, then K(X, Z) also has property SU in L(X, Z). If X is a Banach space such that $X^{*}$ has the metric compact approximation property with adjoint operators, then M-ideal (resp. HB-subspace) property of K(X, Y) in L(X, Y) is inherited to K(X, Z) in L(X, Z).

M-아이디얼의 기원과 그의 신비성

  • 조총만
    • Journal for History of Mathematics
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    • v.13 no.2
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    • pp.41-48
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    • 2000
  • In this paper, we explore the origin of the notion of an M-ideal, and appreciate the richness of algebraic and geometric properties of an M-ideal.

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M-IDEALS AND PROPERTY SU

  • Cho, Chong-Man;Roh, Woo-Suk
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.4
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    • pp.663-668
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    • 2001
  • X and Y are Banach spaces for which K(X, Y), the space of compact operators from X to Y, is an M-ideal in L(X, Y), the space of bounded linear operators form X to Y. If Z is a closed subspace of Y such that L(X, Z) has property SU in L(X, Y) and d(T, K(X, Z)) = d(T, K(X, Y)) for all $T \in L(X, Z)$, then K(X, Z) is an M-ideal in L(X, Z) if and only if it has property SU is L(X, Z).

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Remarks on M-ideals of compact operators

  • Cho, Chong-Man
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.445-453
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    • 1996
  • A closed subspace J of a Banach space X is called an M-ideal in X if the annihilator $J^\perp$ of J is an L-summand of $X^*$. That is, there exists a closed subspace J' of $X^*$ such that $X^* = J^\perp \oplus J'$ and $\left\$\mid$ p + q \right\$\mid$ = \left\$\mid$ p \right\$\mid$ + \left\$\mid$ q \right\$\mid$$ wherever $p \in J^\perp and q \in J'$.

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A NOTE ON M-IDEALS OF COMPACT OPERATORS

  • Cho, Chong-Man;Kim, Beom-Sool
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.683-687
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    • 1998
  • Suppose X is a subspace of $(\sum_{n=1} ^{\infty} X_n)_{c_0}$, dim $X_n<{\infty}$, which has the metric compact approximation property. It is proved that if Y is a Banach space of cotype q for some $2{\leq}1<{\infty}$ then K(X,Y) is an M-ideal in L(X,Y).

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The metric approximation property and intersection properties of balls

  • Cho, Chong-Man
    • Journal of the Korean Mathematical Society
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    • v.31 no.3
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    • pp.467-475
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    • 1994
  • In 1983 Harmand and Lima [5] proved that if X is a Banach space for which K(X), the space of compact linear operators on X, is an M-ideal in L(X), the space of bounded linear operators on X, then it has the metric compact approximation property. A strong converse of the above result holds if X is a closed subspace of either $\elll_p(1 < p < \infty) or c_0 [2,15]$. In 1979 J. Johnson [7] actually proved that if X is a Banach space with the metric compact approximation property, then the annihilator K(X)^\bot$ of K(X) in $L(X)^*$ is the kernel of a norm-one projection in $L(X)^*$, which is the case if K(X) is an M-ideal in L(X).

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