A derivative-Hilbert operator acting on Dirichlet spaces

Abstract

Abstract Let μ \mu be a positive Borel measure on the interval [ 0 , 1 ) \left[0,1) . The Hankel matrix H μ = ( μ n , k ) n , k ≥ 0 {{\mathcal{ {\mathcal H} }}}_{\mu }={\left({\mu }_{n,k})}_{n,k\ge 0} with entries μ n , k = μ n + k {\mu }_{n,k}={\mu }_{n+k} , where μ n = ∫ [ 0 , 1 ) t n d μ ( t ) {\mu }_{n}={\int }_{\left[0,1)}{t}^{n}{\rm{d}}\mu \left(t) , induces formally the operator as follows: DH μ ( f ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n , k a k ( n + 1 ) z n , z ∈ D , {{\mathcal{D {\mathcal H} }}}_{\mu }(f)\left(z)=\mathop{\sum }\limits_{n=0}^{\infty }\left(\mathop{\sum }\limits_{k=0}^{\infty }{\mu }_{n,k}{a}_{k}\right)\left(n+1){z}^{n},\hspace{1em}z\in {\mathbb{D}}, where f ( z ) = ∑ n = 0 ∞ a n z n f\left(z)={\sum }_{n=0}^{\infty }{a}_{n}{z}^{n} is an analytic function in D {\mathbb{D}} . In this article, we characterize those positive Borel measures on [ 0 , 1 ) \left[0,1) for which DH μ {{\mathcal{D {\mathcal H} }}}_{\mu } is bounded (resp. compact) from Dirichlet spaces D α ( 0 < α ≤ 2 ) {{\mathcal{D}}}_{\alpha }\hspace{0.33em}\left(0\lt \alpha \le 2) into D β ( 2 ≤ β < 4 ) {{\mathcal{D}}}_{\beta }\hspace{0.33em}\left(2\le \beta \lt 4) .