  It is not usual to characterize an operator valued series via completeness of
multiplier spaces. In this study, by using a series of bounded linear
operators, we introduce the space $M^\infty_{R}\big(\sum_k T_k\big)$ of Riesz
summability which is a generalization of the Ces\`{a}ro summability. Therefore,
we give the completeness criteria of these spaces with $c_0(X)$-multiplier
convergent operator series. It is a natural consequence that one can
characterize the completeness of a normed space through
$M^\infty_{R}\big(\sum_k T_k\big)$ which will be assumed that is complete for
every $c_0(X)$-multiplier Cauchy operator series. Then, we characterize the
continuity and the (weakly) compactness of the summing operator $\mathcal{S}$
from the multiplier space $M^\infty_{R}\big(\sum_k T_k\big)$ to an arbitrary
normed space $Y$ through $c_0(X)$-multiplier Cauchy and
$\ell_\infty(X)$-multiplier convergent series, respectively. We also prove that
if $\sum_kT_k$ is $\ell_\infty(X)$-multiplier Cauchy, then the multiplier space
of weakly Riesz-convergence associated to the operator valued series
$M^\infty_{wR}\big(\sum_k T_k\big)$ is subspace of $M^\infty_{R}\big(\sum_k
T_k\big)$. Among other results, finally, we obtain a new version of the
well-known Orlicz-Pettis theorem by using Riesz-summability.
