Let D be the unit disc in C. If μ is a finite positive Borel measure on the interval [0, 1) and f is an analytic function in D, f(z)=∑∞n=0anzn (z∈D), we define Cμ(f)(z)=∑n=0∞μn(∑k=0nak)zn,z∈D,
where, for n≥0, μn denotes the n-th moment of the measure μ, that is, μn=∫[0,1)tndμ(t). In this way, Cμ becomes a linear operator defined on the space Hol(D) of all analytic functions in D. We study the action of the operators Cμ on distinct spaces of analytic functions in D, such as the Hardy spaces Hp, the weighted Bergman spaces Apα, BMOA, and the Bloch space B.
