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Compact operators whose adjoints factor through subspaces of $l_{p}$

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For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if $ K ⊂ {∑_{n=1}^{∞} αₙxₙ: {αₙ} ∈ Ball(l_{p'})}$, where p' = p/(p-1) and ${xₙ} ∈ l_{p}^{s}(X)$. An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering ${xₙ} ∈ l_{p}^{w}(X)$. It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of $l_{p}$ in a particular manner. The normed operator ideals $(K_{p},κ_{p})$ of p-compact operators and $(W_{p},ω_{p})$ of weakly p-compact operators, arising from these factorizations, are shown to be complete. It is also shown that the adjoints of p-compact operators are p-summing.
It is further proved that for p ≥ 1 the identity operator on X can be approximated uniformly on every p-compact set by finite rank operators, or in other words, X has the p-approximation property, if and only if for every Banach space Y the set of finite rank operators is dense in the ideal $K_{p}(Y,X)$ of p-compact operators in the factorization norm $ω_{p}$. It is also proved that every Banach space has the 2-approximation property while for each p > 2 there is a Banach space that fails the p-approximation property.
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  • Department of Mathematics, Dyal Singh College, University of Delhi, Lodi Road, New Delhi 110 003, India
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  • Department of Mathematics, Deen Dayal Upadhyaya College, University of Delhi, Karam Pura, New Delhi 110 015, India
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-1-3
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