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 k i=1 I (Ei ) is closed in H∞. © 2010 Elsevier Inc. All rights reserved