This is an announcement for the paper "Compactness and an approximation property related to an operator ideal" by Anil Kumar Karn and Deba Prasad Sinha.
Abstract: For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$, ${\mathcal K}\circ{\mathcal A}$ and ${\mathcal K}\circ{\mathcal A}\circ{\mathcal K}$, where $\mathcal K$ is the ideal of compact operators. We introduce a notion of an $\mathcal A$-approximation property on a Banach space and characterise it in terms of the density of finite rank operators in ${\mathcal A}\circ{\mathcal K}$ and ${\mathcal K}\circ{\mathcal A}$. We propose the notions of $\ell _{\infty}$-extension and $\ell_1$-lifting properties for an operator ideal $\mathcal A$ and study ${\mathcal A}\circ{\mathcal K}$, ${\mathcal }\circ{\mathcal A}$ and the $\mathcal A$-approximation property where $\mathcal A$ is injective or surjective and/or with the $\ell _{\infty}$-extension or $\ell _1$-lifting property. In particular, we show that if $\mathcal A$ is an injective operator ideal with the $\ell _\infty$-extension property, then we have: {\item{(a)} $X$ has the $\mathcal A$-approximation property if and only if $({\mathcal A}^{min})^{inj}(Y,X)={\mathcal A}^{min}(Y,X)$, for all Banach spaces $Y$. \item{(b)} The dual space $X^*$ has the $\mathcal A$-approximation property if and only if $(({\mathcal A}^{dual})^{min})^{sur}(X,Y)=({\mathcal A}^{dual})^{min}(X,Y)$, for all Banach spaces $Y$.}For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$,
Archive classification: math.FA
Mathematics Subject Classification: Primary 46B50, Secondary 46B20, 46B28, 47B07
Remarks: 23 pages
Submitted from: anilkarn@niser.ac.in
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1207.1947
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