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

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...

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Bibliographic Details
Published in:arXiv.org 2022-03
Main Author: Reinov, Oleg
Format: Article
Language:English
Subjects:
Group theory
Hilbert space
Operators
Online Access:Get full text
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Summary: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.
ISSN:2331-8422