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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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