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Riesz multipliers on \(L_ 1(G)\). (English) Zbl 0646.43002
Let X be a complex Banach space, B(X) the Banach space of all bounded linear operators on X. An operator \(A\in B(X)\) is said to be a Fredholm operator iff dim(Ker A) and codim(range A) are both finite. \(A\in B(X)\) is called a Riesz operator iff \(\lambda\) I-A is a Fredholm operator for each \(\lambda\neq 0.\)
Let G be a compact abelian group, \(\mu\) a bounded regular complex Borel measure on G inducing the multiplier \(T_{\mu}\) on \(L_ 1(G)\). The author proves that \(T_{\mu}\) is a Riesz operator iff there exists some positive integer n such that \(\mu^ n\) is absolutely continuous with respect to the Haar measure in G.
Reviewer: G.Crombez

MSC:
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A75 Harmonic analysis on specific compact groups
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