A Derivative-hilbert Operator On A Class Of Analytic Function Space

Suppose that μ is a positive Borel measure on the interval[0,1),and let Hμbe the matrix(μn,k)n,k≥0.The matrix acts on the space of all analytic functions in the unit disk by multiplication on Taylor coefficients,and formally induces the operator Hμ(f)(z)=∑m=0∞(∑k=0∞μn,kak)an.The Hankel matrix Hμ=(μn,k)n,k≥0 with entries μn,k=μn+k,where μn=∫[0,1)tndμ(t),formally induces the operator (?)where f(z)=∑n=0∞anzn is an analytic function in D.We prove the boundedness and compactness of the Derivative-Hilbert operator on a class of analytic function spaces in this thesis.In the first part of this thesis,we mainly discuss the conditions such that the derivative-Hilbert operator DHμ is well defined in the Dirichlet space.Next,we characterize the measure μ for which DHμ is a bounded(resp.compact)operator from Dα into Dβ by using the series expressions,where 0<α≤2,2≤β<4.In the second part of this thesis,the well known fact that the conditions such that the derivative-Hilbert operator DHμ is well defined in the Bergman space.Next,we characterize the measure μ for which DHμ is a bounded(resp.compact)operator from Ap into Hq by using the integral transformation,where 0<p<∞and 0<q<∞.In the third part of this thesis,we obtain the conditions such that the derivative-Hilbert operator DHμ is well defined in the logarithmic Bloch space.Next,we characterize the measure μ for which DHμ is a bounded(resp.compact)operator from BLα into Ap(B),where 0≤α<α.