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

Remarks on some results of G. Pisier and P. Saab on convolutions