A Derivative-Hilbert operator acting on Bergman spaces

Abstract

Let Hμ be the Hankel matrix with entries μn,k=∫[0,1)(n+1)tn+kdμ(t), where μ is a positive Borel measure on the interval [0,1). The matrix acts on the space of all analytic functions in the unit disk by multiplication on Taylor coefficients and induces formally the operatorDHμ(f)(z)=∑n=0∞(∑k=0∞μn,kak)zn, where f(z)=∑n=0∞anzn is an analytic function in D. In this paper, we characterize the measures μ for which DHμ is a bounded (resp., compact) operator from the Bergman space Ap (0<p<∞) into the space Aq (q≥p and q>1), or from Ap (0<p≤1) into A1.