Generalized Hilbert operators acting from Hardy spaces to weighted Bergman spaces

Let $μ$ be a positive Borel measure on the interval $[0,1)$. For $α>0$, the generalized Hankel matrix Hμ, α=(μn, k, α)n, k ≥ 0 with entries μn, k, α=∫[0,1) Γ(n+α)/n ! Γ(α) tⁿ⁺ᵏ dμ(t) induces formally the operator {equation*} Hμ, α(f)(z)=∑ₙ₌₀∞(∑ₖ₌₀∞ μn, k, α a_k) z^n {equation*} on the space of all analytic function f(z)=∑ₖ₌₀∞ aₖ zᵏ in the unit disk D. In this paper, we characterize the measures $μ$ for which Hμ, α(f) is well defined on the Hardy spaces Hᵖ(0<p<∞) and satisfies Hμ, α(f)(z)=∫[0,1) f(t)/(1-t z)^α d μ(t). Among these measures, we further describe those for which Hμ, α(α>1) is a bounded (resp., compact) operator from the Hardy spaces Hᵖ(0<p<∞) into the weighted Bergman spaces Aα-2q.