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

Let $A$ be a Banach space, $p>1$, and $1\mathrm{/}p+1\mathrm{/}q=1$. If a sequence $a=(a_i)$ in $A$ has a finite $p$-sum, then the operator ${\mathrm{\Lambda }}_a:{\mathrm{\ell }}^q\to A$, defined by ${\mathrm{\Lambda }}_a(\beta )={\sum }_{i=1}^{\mathrm{\infty }}{\beta }_i{a}_i,\beta =({\beta }_i)\in {\mathrm{\ell }}^q$, is compact. We present a characterization of compact operators $\mathrm{\Lambda }:{\mathrm{\ell }}^q\to A$, and prove that $\mathrm{\Lambda }$ is compact if and only if $\mathrm{\Lambda }={\mathrm{\Lambda }}_a$, for some sequence $a=(a_i)$ in $A$ with $\{(\phi (a_i)):\phi \in A^{\ast },\mathrm{\parallel }\phi \mathrm{\parallel }\le 1\}$ being a totally bounded set in ${\mathrm{\ell }}^p$. For a sequence $(T_i)$ of bounded operators on a Hilbert space $H$, the corresponding operator $T:{\mathrm{\ell }}^q\to B(H)$, defined by $T(\beta )={\sum }_{i=1}^{\mathrm{\infty }}{\beta }_i{T}_i$, is compact if and only if the set $\{⟨Tx,x⟩:\mathrm{\parallel }x\mathrm{\parallel }=1\}$ is a totally bounded subset of ${\mathrm{\ell }}^p$, where $⟨Tx,x⟩=(⟨T_{1}x,x⟩,⟨T_{2}x,x\text{rangel},\dots )$, for $x\in H$. Similar results are established for $p=1$ and $p=\mathrm{\infty }$.