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Girela-Álvarez, Daniel Merchán-Álvarez, Noel 2024-01-25T08:18:47Z 2024-01-25T08:18:47Z 2018-12-03 Girela, D., Merchán, N. Hankel matrices acting on the Hardy space and on Dirichlet spaces. Rev Mat Complut 32, 799–822 (2019). https://doi.org/10.1007/s13163-018-0288-z https://hdl.handle.net/10630/29170 10.1007/s13163-018-0288-z 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. eng open access Hankel, Operadores de Hilbert, Operadores en espacio de Álgebra lineal Hankel matrices acting on the Hardy space H1 and on Dirichlet spaces. journal article