Composition of analytic paraproducts

For a fixed analytic function g on the unit disc D, we consider the analytic paraproducts induced by g, which are defined by T_gf(z)=∫₀^zf(ζ)g^′(ζ)dζ, S_gf(z)=∫₀^zf^′(ζ)g(ζ)dζ, and M_gf(z)=f(z)g(z). The boundedness of these operators on various spaces of analytic functions on D is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example T_g²,T_gS_g,M_gT_g, etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol g. In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol g than the case of a single paraproduct.