Factorization of Blaschke products and primary ideals in H∞

Let E be a compact subset of G, the union set of nontrivial Gleason parts, and I(E) be the associate primary ideal of H∞. We give a characterization of the numbering function ord(I(E),x), the zero's order of I(E) at x in E, using the geometrical words of E. We also give some factorization theorems of Blaschke products. Using these, we give some descriptions of the higher order hulls of I(E), and for a Carleson–Newman Blaschke product in I(E) it is proved that I(E) is generated by its subproducts as a closed ideal. When each Ei is ρ-separated, we also prove that the tensor product is closed in H∞.