Advanced Search

Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator ${}_{a,b}$ by ${}_{a,b}\left(f\right)\left(z\right)=\Gamma (a+1)/\Gamma (b+1){\int }_0^1\left(f\left(t\right){(1-t)}^b\right)/\left({(1-tz)}^{a+1}\right)dt$, where a and b are non-negative real numbers. In particular, for a = b = β, ${}_{a,b}$ becomes the generalized Hilbert operator ${}_{\beta }$, and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that ${}_{a,b}$ is bounded on Dirichlet-type spaces $S^p$, 0 < p < 2, and on Bergman spaces $A^p$, 2 < p < ∞. Also we find an upper bound for the norm of the operator ${}_{a,b}$. These generalize...