A generalized Hilbert matrix acting on Hardy spaces

If μ is a positive Borel measure on the interval [ 0 , 1 ) , the Hankel matrix H μ = ( μ n , k ) n , k ⩾ 0 with entries μ n , k = ∫ [ 0 , 1 ) t n + k d μ ( t ) induces formally the operator H μ ( f ) ( z ) = ∑ n = 0 ∞ ( ∑ k = 0 ∞ μ n , k a k ) z n on the space of all analytic functions f ( z ) = ∑ k = 0 ∞ a k z k , in the unit disc D . In this paper we describe those measures μ for which H μ is a bounded (compact) operator from H p into H q , 0 < p , q < ∞ . We also characterize the measures μ for which H μ lies in the Schatten class S p ( H 2 ) , 1 < p < ∞ .