  A result of G. Pisier says that a convolution operator $\star f : M(G) \to
C(G),$ where $G$ is a compact Abelian group, can be factored through a Hilbert
space if and only if $f$ has the absolutely summable set of Fourier
coefficients. P. Saab (2010) generalized this result in some directions in the
vector-valued cases. We give some further generalizations of the results of G.
Pisier and P. Saab, considering, in particular, the factorizations of the
operators through the operators of Schatten classes in Hilbert spaces. Also,
some related theorem on the factorization of operators through the operators of
the Lorentz-Schatten classes are obtained.
