FINITE BLASCHKE PRODUCTS AS COMPOSITIONS OF OTHER FINITE

BLASCHKE PRODUCTS

CARL C. COWEN

Abstract. These notes answer the question “When can a finite Blaschke product B be written as a

composition of two finite Blaschke products B1 and B2, that is, B = B1 ◦ B2, in a non-trivial way, that

is, where the order of each is greater than 1.” It is shown that a group can be computed from B and its

local inverses, and that compositional factorizations correspond to normal subgroups of this group. This

manuscript was written in 1974 but not published because it was pointed out to the author that this was

primarily a reconstruction of work of Ritt from 1922 and 1923, who reported on work on polynomials. It

is being made public now because of recent interest in this subject by several mathematicians interested

in different aspects of the problem and interested in applying these ideas to complex analysis and operator

theory.

1. Introduction

From the point of view of these notes, for a positive integer n, a Blaschke product of order n (or n-foldBlaschke product) is an n-to-one analytic map of the open unit disk, D, onto itself. It is well known thatsuch maps are rational functions of order n, so have continuous extensions to the closed unit disk and theRiemann sphere that are n-to-one maps of these sets onto themselves, and have the form

B(z) = λn�

k=1

z − αj

1− αjz

for α1, α2, · · · , αn points of the unit disk and |λ| = 1.These notes were my first formal mathematical writing, developed at the beginning of the work on my

thesis, and were written as a present to my former teacher Professor John Yarnelle on the occasion of hisretirement from Hanover College, Hanover, Indiana, where I had been a student. The only original of thesenotes was given to Professor Yarnelle (since deceased) in December 1974 and what is presented here is ascan of the XeroxTM copy I made for myself at that time. These notes have never been formally circulated,but they have been shared over the years with several people and form the basis of the work in my thesis [2],especially in Section 2, my further work on commutants of analytic Toeplitz operators then [3, 4, 5], andmore recently in work on multiple valued composition operators [6] and a return to questions of commutantsof analytic multiplication operators [7]. In addition, they have formed the basis of my talk “An UnexpectedGroup”, given to several undergraduate audiences in recent years starting in 2007 at Wabash College. Inthe past few years, more interest has been shown in this topic and it seems appropriate to make these ideaspublic and available to others who are working with related topics. Examples of a revival of interest ofRitt’s ideas are in the work of R. G. Douglas and D. Zheng and their collaborators, for example [9, 10], andin the purely function theoretic questions such as the very nice work of Rickards [12] on decomposition ofpolynomials and the paper of Beardon and Ng [1].

The reason this is the first time these notes are being circulated is simple. In the fall of 1976, I gave atalk on this work in the analysis seminar at the University of Illinois at Urbana-Champaign where I wasmost junior of postdocs. The audience received it politely, and possibly with some interest, so as the endof the talk neared, I was feeling good at my first foray into departmental life. Then, at question time, the

Date: Fall 1974; 16 July 2012.

2010 Mathematics Subject Classification. Primary: 30D05; Secondary: 30J10, 26C15, 58D19.

Key words and phrases. Blaschke product, rational function, composition, group action.

1

2 CARL C. COWEN

very distinguished Professor Joe Doob asked, “Didn’t Ritt [13, 14] do something like this in the 1920’s?” Iwas devastated and embarrassed and promptly put the manuscript in a drawer, thinking it unpublishable.In retrospect, I probably should have gone to Professor Doob for advice and written it up for publicationwith appropriate citations. Because Ritt’s first work was on compositional factorization of polynomials, itis somewhat different than this, but it is obvious that the ideas involved apply to polynomials, Blaschkeproducts, or rational functions more generally. I believe that many analysts today, as I was then, are ignorantof Ritt’s work in this arena and, at the very least, his work deserves to be better known.

The remainder of this document is the scan of the original work for Professor Yarnelle from Fall 1974, ashort addendum from a year later that describes the application of these ideas to factorization of an analyticmap on the disk into a composition of an analytic function and a finite Blaschke product, and a shortbibliography of some work related to these ideas.

In the original notes, given a (normalized) finite Blaschke product I of order n, a group GI is describedas a permutation group of the branches of the local inverses of the Blaschke product I acted on by loops(based at 0) in a subset of the disk, the disk with n(n− 1) points removed. The main theorem of the notesis the following.

Theorem 3.1. Let I be a finite Blaschke product normalized as above.If P is a partition of the set of branches of I−1 at 0, {g1, g2, · · · , gn}, that GI respects, then there are

finite Blaschke products JP and bP with the order of bP the same as the order of P so that

I = JP ◦ bPConversely, if J and b are finite Blaschke products so that I = J ◦ b, then there is a partition Pb of the

set of branches of I−1 at 0 which GI respects such that the order of Pb is the same as the order of b.Moreover, if P and b are as above, then

PbP = Pb and bPb = b

It is shown that the compositional factorizations of GI are associated with normal subgroups of GI ,but that the association is more complicated than one might hope in that non-trivial normal subgroupsof GI can be associated with trivial compositional factorizations of I. However, the association is strongenough, then if one knows all of the normal subgroups of GI , then one can construct all possible non-trivial factorizations of I into compositions of finite Blaschke products and inequivalent factorizations of Ias compositions correspond to different normal subgroups of GI .

The main theorem of the addendum is the following.

Theorem. If f : D �→ f(D) is analytic and exactly n-to-one [as a map of the open unit disk onto the image

f(D)], then there is a finite Blaschke product φ and a one-to-one function �f so that f = �f ◦ φ.

This result has the obvious corollaries that f(D) is simply connected and f � has exactly n − 1 zeros inthe disk.

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References

[1] A.F. Beardon and T.W. Ng, On Ritt’s factorization of polynomials, J. London Math. Soc. 62(2000),127–138.

[2] C.C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239(1978),1–31.

[3] C.C. Cowen, The commutant of an analytic Toeplitz operator with automorphic symbol, in HilbertSpace Operators, Lecture Notes in Math. 693, Springer-Verlag, Berlin, 1978, 71-75.

[4] C.C. Cowen, The commutant of an analytic Toeplitz operator, II, Indiana Univ. Math. J. 29(1980),1–12.

[5] C.C. Cowen, An analytic Toeplitz operator that commutes with a compact operator, J. FunctionalAnalysis 36(1980), 169–184.

[6] C.C. Cowen and E.A. Gallardo-Gutierrez, A new class of operators and a description of adjointsof composition operators, J. Functional Analysis 238(2006), 447–462.

[7] C.C. Cowen and R.G. Wahl, Some old thoughts about commutants of analytic multiplicationoperators, preprint, 2012.

[8] J.A. Deddens and T.K. Wong, The commutant of analytic Toeplitz operators,Trans. Amer. Math. Soc. 184(1973), 261–273.

[9] R.G. Douglas, M. Putinar, and K. Wang, Reducing subspaces for analytic multipliers of theBergman space, preprint, 2011.

[10] R.G. Douglas, S. Sun, and D. Zheng, Multiplication operators on the Bergman space via analyticcontinuation, Advances in Math. 226(2011), 541–583.

[11] R. McLaughlin, Exceptional sets for inner functions, J. London Math. Soc. 4(1972), 696–700.[12] J. Rickards, When is a polynomial a composition of other polynomials?, Amer. Math. Monthly

118(2011), 358–363.[13] J.F. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23(1922), 51–66.[14] J.F. Ritt, Permutable rational functions, Trans. Amer. Math. Soc. 25(1923), 399–448.[15] S. Saks and A. Zygmund, Analytic Functions, PWN, Warszawa, 1965.[16] J.E. Thomson, Intersections of commutants of analytic Toeplitz operators, Proc. Amer. Math. Soc.

52(1975), 305–310.[17] B.L. van der Waerden, Modern Algebra, Fredrick Ungar Publishing, New York, 1953.

IUPUI (Indiana University – Purdue University, Indianapolis), Indianapolis, Indiana 46202-3216E-mail address: ccowen@math.iupui.edu