A derivative-Hilbert operator acting on Dirichlet spaces

Abstract Let μ \mu be a positive Borel measure on the interval [ 0 , 1 ) \left[0,1) . The Hankel matrix H = ( n k ≥ {{\mathcal{ {\mathcal H} }}}_{\mu }={\left({\mu }_{n,k})}_{n,k\ge 0} with entries + {\mu }_{n,k}={\mu }_{n+k} , where ∫ t mathvariant="normal">d }_{n}={\int }_{\left[0,1)}{t}^{n}{\rm{d}}\mu \left(t) induces formally operator as follows: xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> mathvariant="script">DH f z ∑ ∞ a ∈ mathvariant="double-struck">D {{\mathcal{D }(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}}, f\left(z)={\sum }_{n=0}^{\infty }{a}_{n}{z}^{n} is an analytic function in {\mathbb{D}} In this article, we characterize those measures for which } bounded (resp. compact) from Dirichlet spaces mathvariant="script">D α width="0.33em" < ≤ 2 {{\mathcal{D}}}_{\alpha }\hspace{0.33em}\left(0\lt \alpha \le 2) into β 4 {{\mathcal{D}}}_{\beta }\hspace{0.33em}\left(2\le \beta \lt 4)