Generalized Hilbert operators acting on weighted Bergman spaces and Dirichlet spaces

Let μ be a positive Borel measure on the interval [0, 1). For β > 0 , the generalized Hankel matrix H μ , β = ( μ n , k , β ) n , k ≥ 0 with entries μ n , k , β = ∫ [ 0.1 ) Γ ( n + β ) n ! Γ ( β ) t n + k d μ ( t ) induces formally the operator H μ , β ( f ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n , k , β a k z n , on the space of all analytic function f ( z ) = ∑ k = 0 ∞ a k z n in the unit disk D . In this paper, we characterize those positive Borel measures on [0, 1) such that H μ , β ( f ) ( z ) = ∫ [ 0 , 1 ) f ( t ) ( 1 - t z ) β d μ ( t ) for all f in the weighted Bergman spaces A α p ( 0 < p < ∞ , α > - 1 ) , and among them, we describe those for which H μ , β ( β > 0 ) is a bounded (resp., compact) operator on weighted Bergman spaces A α p and Dirichlet spaces D α p .