  We investigate convolution operators in the sequence spaces $d_p$, for $1\le
p<\infty$. These spaces, for $p>1$, arise as dual spaces of the \ces sequence
spaces $ces_p$ thoroughly investigated by G.~Bennett. A detailed study is also
made of the algebra of those sequences which convolve $d_p$ into $d_p$. It
turns out that such multiplier spaces exhibit features which are very different
to the classical multiplier spaces of $\ell^p$.
