Geodesic
Regular factorization of the scalar characteristic function
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 19-50

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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. 
@article{ZNSL_2000_270_a1,
     author = {M. F. Gamal'},
     title = {Regular factorization of the scalar characteristic function},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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     publisher = {mathdoc},
     volume = {270},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a1/}
}
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M. F. Gamal'. Regular factorization of the scalar characteristic function. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 19-50. http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a1/