Compactness and an approximation property related to an operator ideal

2012/07/09 by Karn, Anil Kumar, Sinha, Deba Prasad
#46B28 #46B50 (Primary) 46B20 #47B07 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1207.1947

Abstract

For an operator ideal \mathcal A, we study the composition operator ideals \mathcal A∘\mathcal K, \mathcal K∘\mathcal A and \mathcal K∘\mathcal A∘\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∘\mathcal K and \mathcal K∘\mathcal A. We propose the notions of ℓ_∞-extension and ℓ₁-lifting properties for an operator ideal \mathcal A and study \mathcal A∘\mathcal K, \mathcal∘\mathcal A and the \mathcal A-approximation property where \mathcal A is injective or surjective and/or with the ℓ_∞-extension or ℓ₁-lifting property. In particular, we show that if \mathcal A is an injective operator ideal with the ℓ_∞-extension property, then we have: (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. (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∘\mathcal K,

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