A characterization of compact operators on $\ell^p$-spaces

Abstract

Let $A$ be a Banach space, $p>1$, and $1/p+1/q=1$. If a sequence $a=(a_i)$ in $A$ has a finite $p$-sum, then the operator $\Lambda_a:\ell^q\to A$, defined by $\Lambda_a(\beta)=\sum_{i=1}^\infty \beta_i a_i, \beta=(\beta_i)\in \ell^q$, is compact. We present a characterization of compact operators $\Lambda:\ell^q\to A$, and prove that $\Lambda$ is compact if and only if $\Lambda=\Lambda_a$, for some sequence $a=(a_i)$ in $A$ with $\{(\phi(a_i)): \phi\in A^*, \|\phi\|\leq 1\}$ being a totally bounded set in $\ell^p$. For a sequence $(T_i)$ of bounded operators on a Hilbert space $H$, the corresponding operator $T:\ell^q\to B(H)$, defined by $T(\beta) = \sum_{i=1}^\infty \beta_i T_i$, is compact if and only if the set $\{\langle T x,x\rangle:\|x\|=1\}$ is a totally bounded subset of $\ell^p$, where $\langle T x,x\rangle = (\langle T_1 x,x\rangle, \langle T_2 x,x\rangel, \dotsc)$, for $x\in H$. Similar results are established for $p=1$ and $p=\infty$.