Compact operators whose adjoints factor through subspaces of $l_p$
Volume 150 / 2002
Abstract
For $p\geq 1$, a subset $K$ of a Banach space $X$ is said to be relatively $p$-compact if $ K \subset \{ \sum _{n=1}^\infty {\alpha_n x_n} : \{\alpha_n\} \in \mathop{\rm Ball}\nolimits (l_{p'}) \}, $ where $p'={p/(p-1)}$ and $\{x_n\}\in l_p^{\rm s}(X)$. An operator $T \in B(X,Y)$ is said to be $p$-compact if $T(\mathop{\rm Ball}\nolimits (X))$ is relatively $p$-compact in $Y$. Similarly, weak $p$-compactness may be defined by considering $\{x_n\}\in l_p^{\rm w}(X)$. 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 $(K_p,\kappa _p)$ of $p$-compact operators and $(W_p, \omega_p)$ 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 $p\geq 1$ 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 $\omega_p$. 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.
