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http://dx.doi.org/10.4134/BKMS.2004.41.3.457

HEREDITARY PROPERTIES OF CERTAIN IDEALS OF COMPACT OPERATORS  

Cho, Chong-Man (Department of Mathematics, Hanyang University)
Lee, Eun-Joo (Department of Mathematics, Hanyang University)
Publication Information
Bulletin of the Korean Mathematical Society / v.41, no.3, 2004 , pp. 457-464 More about this Journal
Abstract
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).
Keywords
ideal; M-ideal; H B-subspace; property SU; compact operator;
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