Finite Blaschke Products and Their Connections PDF
Preview Finite Blaschke Products and Their Connections
Stephan Ramon Garcia Javad Mashreghi · William T. Ross Finite Blaschke Products and Their Connections Finite Blaschke Products and Their Connections Stephan Ramon Garcia • Javad Mashreghi William T. Ross Finite Blaschke Products and Their Connections 123 StephanRamonGarcia JavadMashreghi DepartmentofMathematics DepartmentofMathematicsandStatistics PomonaCollege LavalUniversity Claremont,CA,USA QuébecCity,QC,Canada WilliamT.Ross DepartmentofMathematicsandComputer Science UniversityofRichmond Richmond,VA,USA ISBN978-3-319-78246-1 ISBN978-3-319-78247-8 (eBook) https://doi.org/10.1007/978-3-319-78247-8 LibraryofCongressControlNumber:2018937693 MathematicsSubjectClassification(2010):30J10,30J05,30A78,30A76,30A08,30C15,30E10,47A12, 47N50,47A30 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Toourfamilies: Gizem;Reyhan,andAltay Shahzad;Dorsa,Parisa,andGolsa Fiona Preface This is a book about a beautiful subject that begins with the topic of Möbius transformations.Indeed,Möbiustransformations az+b z(cid:2)→ cz+d are studied in complex analysis since their mapping properties demonstrate won- derful connections with geometry. These transformations map extended circles to extended circles, enjoy the symmetry principle, come in several types yielding different behavior depending on their fixed point(s), and, through an identification with 2×2 matrices, make connections to group theory and projective geometry. Finite Blaschke products, the focus of this book, are products of certain types of Möbiustransformations,theautomorphismsoftheopenunitdiskD,namely w−z z(cid:2)→ξ , 1−wz where |w| < 1 and |ξ| = 1 are fixed. These products have an uncanny way of appearing in many areas of mathematics such as complex analysis, linear algebra, grouptheory,operatortheory,andsystemstheory.ThisbookcoversfiniteBlaschke products and is designed for advanced undergraduate students, graduate students, andresearcherswhoarefamiliarwithcomplexanalysisbutwhowanttoseemore ofitsconnectionstootherfieldsofmathematics.Muchofthematerialinthisbook is scattered throughout mathematical history, often only appearing in its original language, and some of it has never seen a modern exposition. We gather up these gemsandputthemtogetherasacohesivewhole,takingaleisurelypacethroughthe subjectandleavingplentyoftimeforexpositionandexamples.Thereareplentyof exercisesforthereaderwhonotonlywantstoappreciatethebeautyofthesubject buttogainaworkingknowledgeofitaswell. vii viii Preface Intheearlytwentiethcentury,thestudyofinfiniteproductsoftheform (cid:2) |z | z −z B(z)= k k , z 1−z z k(cid:2)1 k k inwhichz ,z ,...isasequenceinD,wasinitiatedin1915byWilhelmBlaschke 1 2 (1885–1962).Thisproductconv(cid:3)ergesuniformlyoncompactsubsetsofDifandonly ifthezerosequencez satisfies (1−|z |)<∞.TheseBlaschkeproductsare k k(cid:2)1 k analyticonDandhavetheadditionalpropertythattheradiallimitlimr→1−B(reiθ) exists and is of unit modulus for almost every θ ∈ [0,2π). In other words, B is an inner function. Blaschke products have been studied intensely since they were firstintroducedandtheyappearinmanycontextsthroughoutcomplexanalysisand operatortheory. ThisbookisconcernedwithfiniteBlaschkeproducts,inwhichthezerosequence z ,z ,...,z is finite and the product terminates. Although the skeptical reader 1 2 n might think this focus is too narrow, there are many fascinating connections with geometry,complexanalysis,andoperatortheorythatdemandattention. TherearealreadysomeexcellenttextsthatcoverinfiniteBlaschkeproductsand, more generally, inner functions [38, 61]. However, as the reader will see, there are many beautiful theorems involving finite Blaschke products that have no clear analogues in the infinite case. Finite Blaschke products are not often discussed in the standard texts onfunction spaces or complex variables sincethefocus thereis often on inner functions as part of the broader theory of Hardy spaces. This book focuses on finite Blaschke products and the many results that pertain only to the finitecase. The book begins with an exposition of the Schur class S, the set of analytic functions from D to D−, the closure of D, and an introduction to hyperbolic geometry. We develop this material from scratch, assuming only that the reader has had a basic course in complex variables. We characterize the finite Blaschke products in several different ways. First, a rational function is a finite Blaschke productifandonlyifitisoftheform α +α z+···+α zn 0 1 n , αn+αn−1zn−1+···+α0zn in which the numerator is a polynomial whose n roots lie in D. Second, a finite BlaschkeproductmapsDontoD(andtheunitcircleTontoitself)preciselyntimes andatheoremofFatouconfirmsthatthesearetheonlyfunctionsthatarecontinuous onD− andanalyticonDwiththisproperty.Third,eachfiniteBlaschkeproductB satisfies lim |B(z)|=1 |z|→1− Preface ix and another result of Fatou shows that the finite Blaschke products are the only analytic functions onDthatdothis.Whether asrationalfunctions whosedefining polynomials enjoy certain symmetries, as n-to-1 analytic functions on D, or as analytic functions with unimodular boundary values, the finite Blaschke products distinguishthemselvesasspecialelementsoftheSchurclass. The approximation of a given analytic function by well-understood functions froma fixed class isa standard technique incomplex analysis.For example, there arethewell-knownapproximationtheoremsofRunge,Mergelyan,andWeierstrass. We examine a few results of this type that involve finite Blaschke products. More specifically,acelebratedtheoremofCarathéodoryensuresthatanyfunctioninthe Schur class S can be approximated, uniformly on compact subsets of D, by a sequenceoffiniteBlaschkeproducts.Infact,onecaneventaketheapproximating Blaschkeproductstohavesimplezeros.AfterCarathéodory’stheorem,wediscuss Fisher’stheorem,whichsaysthatanyfunctioninS thatextendscontinuouslytoD− can be approximated uniformly on D− by convex combinations of finite Blaschke products. As another example, a theorem of Helson and Sarason states that any continuousfunctionfromTtoTcanbeuniformlyapproximatedbyasequenceof quotientsoffiniteBlaschkeproducts. One might think there is not much to say about the zeros of a finite Blaschke product.Afterall,thelocationofthezerosispartofthedefinition!However,there are some beautiful gems here. The famed Gauss–Lucas theorem asserts that if P (cid:6) is a polynomial, then the zeros of P , the derivative of P, are contained in the convexhullofthezerosofP.Therearetheoremsthatsaythatthezerosofafinite BlaschkeproductBarecontainedintheconvexhullofthesolutionstotheequation B(z) = 1 (or indeed the solutions to B(z) = eiθ for any θ ∈ [0,2π)). Moreover, (cid:6) thehyperbolicanalogueoftheGauss–LucastheoremsaysthatthezerosofB (the criticalpointsofB)arecontainedinthehyperbolicconvexhullofthezerosofB. For Blaschke products of low degree, these results are even more explicit and can bestatedintermsofclassicalgeometryinvolvingellipses.Thereisalsoaresultof HeinswhichsaysthatonecancreateafiniteBlaschkeproductwithanydesiredset ofcriticalpointsinD.Finally,foranalyticfunctionsonD−,onecanstate,interms of finite Blaschke products, a curious converse (the Challener–Rubel theorem) to Rouché’stheorem. Interpolation is another important topic in complex analysis. The most basic result in this direction is the Lagrange interpolation theorem, which guarantees thatfordistinctz ,z ,...,z andanyw ,w ,...,w thereisapolynomialP for 1 2 n 1 2 n which P(z ) = w for all j. The connection finite Blaschke products make with j j interpolationcomesfromPick’stheorem:givendistinctz ,z ,...,z ∈Dandany 1 2 n w ,w ,...,w ∈D,thenthereisanf ∈S forwhichf(z )=w forallj ifand 1 2 n j j onlyifthePickmatrix (cid:4) (cid:5) 1−w w j i 1−z z j i 1(cid:3)i,j(cid:3)n x Preface is positive semidefinite. Furthermore, when the interpolation is possible, it can be donewithafiniteBlaschkeproduct.Amoreinvolvedboundaryinterpolationresult istheCantor–Phelpstheorem(forwhichweprovidetwodistinctproofs,oneabstract andanotherconstructive),whichsaysthatgivendistinctζ ,ζ ,...,ζ ∈Tandany 1 2 n ξ ,ξ ,...,ξ ∈TthereisafiniteBlaschkeproductB withB(ζ )=ξ forallj. 1 2 n j j So far we have discussed finite Blaschke product themselves and their connec- tions to well-studied topics in complex analysis (zeros, critical points, residues, valence, approximation, and interpolation). However, as mentioned earlier, finite Blaschkeproductsappearinmanyotherplaces. (cid:3) Forexample,Bohr’sinequalityassertsthatiff = a zn ∈S,then n(cid:2)0 n (cid:6) |a |rn (cid:2)1, r ∈[0,1]. n 3 n(cid:2)0 The number 1 is optimal and is called the Bohr radius for the Schur class. Using 3 finiteBlaschkeproducts,weexploreaBohr-typeinequalityforsubclassesofSchur functionsthatvanishatcertainpointsofDandfortheSchurclassfunctionswhose first several derivatives vanish at zero. It turns out that the extremal functions for theseextendedBohrproblemsarefiniteBlaschkeproducts. NextwecovertwoconnectionsfiniteBlaschkeproductsmakewithgrouptheory. ForafixedfiniteBlaschkeproductB,considerthesetG ofcontinuousfunctions B u:T→TforwhichB◦u=B.OnecanseethatG isasemigroupunderfunction B composition.AtheoremofChalendarandCassierrevealsthatG isacyclicgroup B andthatonecanidentifyageneratorbyconsideringthepreviouslymentionedn-to-1 mappingpropertiesofB onT.Wealsocover,viaCowen’sunpublishedexposition, an old theorem of Ritt that examines when we can write B as a composition B = C◦D,inwhichCandDarefiniteBlaschkeproducts.Theanswerisintermsofthe monodromygroupofB−1.WealsogiveanequivalentformulationofRitt’stheorem intermsofcertainsubgroupsofG . B FiniteBlaschkeproductsalsomakeconnectionstooperatortheory.Forexample, if T is a contraction on a Hilbert space and B is a finite Blaschke product with n zeros, then B(T) is also a contraction. Moreover, a theorem of Gau and Wu says that(cid:8)B(T)(cid:8)=1ifandonlyif(cid:8)Tn(cid:8)=1.Anotherconnectioniswiththenumerical rangeofanoperator.Thespectralmappingtheoremsaysthatσ(p(T))=p(σ(T)), in which σ(T) is the spectrum of a bounded Hilbert space operator T and p is a polynomial. One may wonder whether or not a similar identity W(p(T)) = p(W(T))holdsforthenumericalrange W(T)={(cid:9)Tx,x(cid:10):(cid:8)x(cid:8)=1}. Althoughthedesiredidentityisnottrueingeneral,therearesomesuitablesubsti- tutes.Infact,HalmosaskedwhetherornotW(T)⊆D−impliesthatW(Tn)⊆D− for every n (cid:3) 1. Progress was made when it was shown that if W(T) ⊆ D− and B is a finite Blaschke product with B(0) = 0, then W(B(T)) ⊆ D−. A theorem Preface xi ofBergerandStampfliextendsthisresultfromfiniteBlaschkeproductsthatvanish at the origin to the Schur functions that are continuous on D− and vanish at the origin. However, without the condition f(0) = 0, there are contractions T with W(T) ⊆ D− for which W(f(T)) intersects the complement of D−. A suitable replacement here is a theorem of Drury which says that though W(f(T)) may intersect the complement of D−, it is contained in a certain “teardrop” region, a slight “bulge” of D. Moreover, the use of finite Blaschke products indicates the sharpnessofDrury’stheorem. StillanotherconnectiontofiniteBlaschkeproductscomeswithmodelsoflinear transformations. In linear algebra, or more broadly in operator theory, one often wants to create a model for certain types of linear transformations. For example, there is the classical spectral theorem from linear algebra which says that any normal matrix is unitarily equivalent to a diagonal matrix. One can show that any contractivematrixT withrank(I−T∗T)=1andwhoseeigenvaluesλ ,λ ,...,λ 1 2 n are contained in D is unitarily equivalent to the compression of the shift operator f (cid:2)→zf ontheHardyspaceH2tothemodelspace (cid:7) (cid:8) 1 span :1(cid:2)j (cid:2)n . 1−λ z j Along with this result, one obtains a function-theoretic characterization of the invariantsubspacesoftheseoperatorsaswell.Infact,thismodelspaceisthevector spaceofrationalfunctionsf withnopolesinD−forwhich (cid:9) 2π dθ f(eiθ)B(eiθ)e−inθ =0, n(cid:3)0, 2π 0 in which B is the finite Blaschke product whose zeros are the eigenvalues λ . j The finite-dimensional approach undertaken in this book is intuitive and prepares interestedreadersforthemoreadvancedtext[59]. FiniteBlaschkeproductscanalsobeusedtoexplorerationalfunctionsf thatare analyticonDandforwhichf(eiθ)isanextendedrealnumberforallθ ∈ [0,2π]. Thesefunctionsaresometimescalledtherealrationalfunctions.Examplesinclude 1+z f(z)=i , 1−z and,moregenerally, B +B f =i 1 2, B −B 1 2 inwhichB andB arefiniteBlaschkeproductssuchthatB −B hasnozeroson 1 2 1 2 D.Infact,atheoremofHelsonsaysthesearealloftherealrationalfunctions.We willdiscussvariouspropertiesofrealrationalfunctionssuchasacharacterizationof
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