Let $\mu$ be a positive Borel measure on the interval [0,1). For $\beta > 0$, The generalized Hankel matrix $\mathcal{ H}_{ \mu,\beta}= (\mu_{ n,k,\beta})_{ n,k\geq0}$ with entries $\mu_{ n,k,\beta}= \int_{ [0.1)}\frac{ \Gamma(n+\beta)}{ n!\Gamma(\beta)} t^{ n+k}d\mu(t)$, induces formally the operator $$\mathcal{ H}_{ \mu,\beta}(f)(z)=\sum_{ n=0}^\infty \left(\sum_{ k=0}^\infty \mu_{ n,k,\beta}a_k\right)z^n$$ on the space of all analytic function $f(z)=\sum_{ k=0}^ \infty a_k z^n$ in the unit disc $\mathbb{ D}$. In this paper, we characterize those positive Borel measures on $[0,1)$ such that $\mathcal{ H}_{ \mu,\beta}(f)(z)= \int_{ [0,1)} \frac{ f(t)}{ { (1-tz)^\beta}} d\mu(t)$ for all in weighted Bergman Spaces $A_{ \alpha}^p(0<p<\infty,\; \alpha>-1)$, and among them we describe those for which $\mathcal{ H}_{ \mu,\beta}(\beta>0)$ is a bounded(resp.,compact) operator on weighted Bergman spaces and Dirichlet spaces.
Categories: math.CV