An harmonic analysis for operators: F. and M. Riesz theorems (Q792569)

The authors consider bounded linear operators on a homogeneous Banach space B on a compact Abelian group G, with ordered dual. Following K. DeLeeuw one associates a formal Fourier series with such operators. When \(B=L^ p(G)\), \(1\leq p<\infty\), or \(B=C(G)\) the authors generalize the F. and M. Riesz theorem stating that a power series on the unit circle cannot vanish on a subset having positive Lebesgue measure without vanishing almost everywhere. For certain homogeneous Banach spaces in this setting the authors generalize also the first F. and M. Riesz theorem stating that a measure whose Fourier transform vanishes for negative indices is absolutely continuous.