Regular factorization of the scalar characteristic function

It is known that the regular factorizations of the characteristic function of an operator describe its invariant subspaces. The case of a scalar characteristic function is considered. Some examples are given. Factorizations describing all chains of invariant subspaces that contain a given subspace $L$ are constructed in terms of the factorization describing $L$. A representation of the regular factorization of a function is obtained in terms of factorizations of its inner and outer parts.