FNTE BLASCHKE PRODUCTS AS COMPOSTONS OF OTHER FNTE 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 B 1 and B 2,thatis,B = B 1 B 2,inanon-trivialway,that is, where the order of each is greater than 1. t 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. t 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. ntroduction From the point of view of these notes, for a positive integer n, a Blaschke product of order n (or n-fold Blaschke product) is an n-to-one analytic map of the open unit disk, D, onto itself. t is well known that such maps are rational functions of order n, so have continuous extensions to the closed unit disk and the Riemann sphere that are n-to-one maps of these sets onto themselves, and have the form B(z) =λ n k=1 z α j 1 α j z for α 1, α 2,, α n 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 his retirement from Hanover College, Hanover, ndiana, where had been a student. The only original of these notes was given to Professor Yarnelle (since deceased) in December 1974 and what is presented here is a scan of the Xerox TM copy 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], and more recently in work on multiple valued composition operators [6] and a return to questions of commutants of analytic multiplication operators [7]. n addition, they have formed the basis of my talk An Unexpected Group, given to several undergraduate audiences in recent years starting in 2007 at Wabash College. n the past few years, more interest has been shown in this topic and it seems appropriate to make these ideas public and available to others who are working with related topics. Examples of a revival of interest of Ritt s ideas are in the work of R. G. Douglas and D. Zheng and their collaborators, for example [9, 10], and in the purely function theoretic questions such as the very nice work of Rickards [12] on decomposition of polynomials and the paper of Beardon and Ng [1]. The reason this is the first time these notes are being circulated is simple. n the fall of 1976, gave a talk on this work in the analysis seminar at the University of llinois at Urbana-Champaign where was most junior of postdocs. The audience received it politely, and possibly with some interest, so as the end of the talk neared, was feeling good at my first foray into departmental life. Then, at question time, the Date: Fall1974;16July2012. 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? was devastated and embarrassed and promptly put the manuscript in a drawer, thinking it unpublishable. n retrospect, probably should have gone to Professor Doob for advice and written it up for publication with appropriate citations. Because Ritt s first work was on compositional factorization of polynomials, it is somewhat different than this, but it is obvious that the ideas involved apply to polynomials, Blaschke products, or rational functions more generally. believe that many analysts today, as was then, are ignorant of 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, a short addendum from a year later that describes the application of these ideas to factorization of an analytic map on the disk into a composition of an analytic function and a finite Blaschke product, and a short bibliography of some work related to these ideas. n the original notes, given a (normalized) finite Blaschke product of order n, a group G is described as a permutation group of the branches of the local inverses of the Blaschke product 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 notes is the following. Theorem 3.1. Let be a finite Blaschke product normalized as above. f P is a partition of the set of branches of 1 at 0, {g 1,g 2,,g n },thatg respects, then there are finite Blaschke products J P and b P with the order of b P the same as the order of P so that = J P b P Conversely, if J and b are finite Blaschke products so that = J b, then there is a partition P b of the set of branches of 1 at 0 which G respects such that the order of P b is the same as the order of b. Moreover, if P and b are as above, then P bp = P b and b Pb = b t is shown that the compositional factorizations of G are associated with normal subgroups of G, but that the association is more complicated than one might hope in that non-trivial normal subgroups of G can be associated with trivial compositional factorizations of. However, the association is strong enough, then if one knows all of the normal subgroups of G, then one can construct all possible nontrivial factorizations of into compositions of finite Blaschke products and inequivalent factorizations of as compositions correspond to different normal subgroups of G. The main theorem of the addendum is the following. Theorem. f 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 in the disk.
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22 CARL C. COWEN 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 Hilbert Space Operators, Lecture Notes in Math. 693, Springer-Verlag, Berlin, 1978, 71-75. [4] C.C. Cowen, The commutant of an analytic Toeplitz operator,, ndiana Univ. Math. J. 29(1980), 1 12. [5] C.C. Cowen, An analytic Toeplitz operator that commutes with a compact operator, J. Functional Analysis 36(1980), 169 184. [6] C.C. Cowen and E.A. Gallardo-Gutiérrez, A new class of operators and a description of adjoints of composition operators, J. Functional Analysis 238(2006), 447 462. [7] C.C. Cowen and R.G. Wahl, Some old thoughts about commutants of analytic multiplication operators, 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 the Bergman space, preprint, 2011. [10] R.G. Douglas, S. Sun, and D. Zheng, Multiplication operators on the Bergman space via analytic continuation, 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, ntersections 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. UPU (ndiana University Purdue University, ndianapolis), ndianapolis, ndiana 46202-3216 E-mail address: ccowen@math.iupui.edu