Compact operators which factor through subspaces of l p

Let 1 ≤ p ≤ ∞. A subset K of a Banach space X is said to be relatively p ‐compact if there is an 〈 x n 〉 ∈ l sp ( X ) such that for every k ∈ K there is an 〈 α n 〉 ∈ l p ′ such that k = σ ∞ n=1α n x n . A linear operator T : X → Y is said to be p ‐compact if T ( Ball ( X )) is relatively p ‐compact in Y . The set of all p ‐compact operators K p ( X, Y ) from X to Y is a Banach space with a suitable factorization norm κ p and ( K p , κ p ) is a Banach operator ideal. In this paper we investigate the dual operator ideal ( K dp , κ dp ). It is shown that κ dp ( T ) = π p ( T ) for all T ∈ B ( X, Y ) if either X or Y is finite‐dimensional. As a consequence it is proved that the adjoint ideal of K dp is I p ′ , the ideal of p ′‐integral operators. Further, a composition/decomposition theorem K dp = Π p K is proved which also yields that (Π minp ) inj = K dp . Finally, we discuss the density of finite rank operators in K dp and give some examples for different values of p in this respect. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)