Factorization of Blaschke products and ideal theory in H∞

Let H∞ be the Banach algebra of bounded analytic functions on the open unit disk D. Let G be the union set of all nontrivial Gleason parts in the maximal ideal space of H∞. Let E be a nonvoid compact and totally disconnected subset of G and nE be a bounded numbering function on E. We characterize nE for which there is a closed ideal I in H∞ such that Z(I)=E and ord(I,x)=nE(x) for every x∈E. Let I1,I2,…,Ik be closed ideals in H∞ satisfying Z(Ii)⊂G for 1⩽i⩽k. We prove that is a closed ideal. A local ideal theory in H∞ plays an important role to prove our results.