Operators whose adjoints are quasi $p$-nuclear
Volume 197 / 2010
Studia Mathematica 197 (2010), 291-304
MSC: 47B07, 46B50, 47B10.
DOI: 10.4064/sm197-3-6
Abstract
For $p\geq 1$, a set $K$ in a Banach space $X$ is said to be relatively $p$-compact if there exists a $p$-summable sequence $(x_n)$ in $X$ with $K\subseteq \{\sum_n\alpha_nx_n : (\alpha_n)\in B_{\ell_{p'}}\}$. We prove that an operator $T\colon X\rightarrow Y$ is $p$-compact (i.e., $T$ maps bounded sets to relatively $p$-compact sets) iff $T^*$ is quasi $p$-nuclear. Further, we characterize $p$-summing operators as those operators whose adjoints map relatively compact sets to relatively $p$-compact sets.
