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 $\{(\varphi (a_i)):\varphi \in A^{\ast },\Vert \varphi \Vert \le 1\}$ 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 $\{⟨Tx,x⟩:\Vert x\Vert =1\}$ 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 $p=\infty$.