A Derivative-Hilbert operator acting on Bergman spaces

Let h(mu) be the Hankel matrix with entries mu n,k = f[0,1)(n + 1)tn+kd mu(t), where mu 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 operator DH mu(f)(z) = Sigma(infinity)(n=0) (Sigma(infinity)(k =0) mu(n),(k)a(k))z(n), where f (z) = E(n=0)(infinity)a(n)z(n) is an analytic function in D. In this paper, we characterize the measures mu for which DH mu is a bounded (resp., compact) operator from the Bergman space Ap (0 < p infinity) into the space A(q) (q >= p and q > 1), or from Ap (0 < p <= 1) into A(1). (C) 2021 Elsevier Inc. All rights reserved.