Discrete Hausdorff transformations

Abstract:

Let $K$ be a complex valued measurable function on $(0,1]$ such that $\int _0^1 {{t^{ - 1/p}}|K(t)|dt}$ is finite for some $p > 1$. Let $H$ be the Hausdorff operator on ${l^p}$ determined by the moments ${\mu _n} = \int _0^1 {{t^n}K(t)} dt$. Define $\Psi (z) = \int _0^1 {{t^z}K(t)} dt$. Then for each $z$ with Re $\operatorname {Re} z > - 1/p,\Psi (z)$ is an eigenvalue of ${H^\ast }$. The spectrum of $H$ is the union of $\{ 0\}$ with the range of $\Psi$ on the half-plane Re $\operatorname {Re} z \geqq - 1/p$.