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

Deba P. Sinha ; Anil K. Karn

Studia Mathematica, Tome 151 (2002), p. 17-33 / Harvested from The Polish Digital Mathematics Library

Résumé

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^{\prime }})}$, 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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Publié le : 2002-01-01

Zbl 1008.46008

EUDML-ID : urn:eudml:doc:286285