Hankel matrices acting on the Hardy space H1 and on Dirichlet spaces.

Abstract

If μ is a positive Borel measure on the interval [0, 1) we let H_μ be the Hankel matrix H_μ={ μ_{n,k} }_{n,k} with entries μ_{n,k} =μ_{n+k} where μ_n denotes the moment of order n of μ. This matrix induces formally an operator on the space of all analytic functions in the unit disc D. When μ is the Lebesgue measure on [0,1) the operator H_μ is the classical Hilbert operator H which is bounded on H^p if 1<p< ∞, but not on H^1. J. Cima has recently proved that H is an injective bounded operator from H^1 into the space C of Cauchy transforms of measures on the unit circle. The operator H_μ is known to be well defined on H^1 if and only if μ is a Carleson measure and in such a case we have that H_μ(H^1) is contained in C. Furthermore, it is bounded from H^1 into itself if and only if μ is a 1-logarithmic 1-Carleson measure. In this paper we prove that when μ is a 1-logarithmic 1-Carleson measure then H_μ actually maps H^1 into the space of Dirichlet type D^1_0. We discuss also the range of H_μ on H^1 when μ is an α-logarithmic 1-Carleson measure (0<α<1). We study also the action of the operators H_μ on Bergman spaces and on Dirichlet spaces.