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

TL;DR

This paper explores advanced factorizations of convolution operators on compact Abelian groups, extending previous results to vector-valued cases and Schatten class operators, revealing new insights into operator theory.

Contribution

It generalizes Pisier and Saab's results by considering factorizations through Schatten and Lorentz-Schatten class operators, broadening the scope of convolution operator analysis.

Findings

Factorization of convolution operators through Schatten class operators.

Extension of previous results to vector-valued and Lorentz-Schatten classes.

New theorems on operator factorizations in Hilbert spaces.

Abstract

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.