Table Of ContentSPRINGER BRIEFS IN MATHEMATICS
Fabrizio Colombo
Irene Sabadini
Daniele C. Struppa
Entire Slice
Regular
Functions
123
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Fabrizio Colombo Irene Sabadini
(cid:129)
Daniele C. Struppa
Entire Slice Regular
Functions
123
Fabrizio Colombo Daniele C.Struppa
Dipartimento di Matematica Schmid Collegeof Science andTechnology
Politecnico di Milano Chapman University
Milan Orange,CA
Italy USA
IreneSabadini
Dipartimento di Matematica
Politecnico di Milano
Milan
Italy
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs inMathematics
ISBN978-3-319-49264-3 ISBN978-3-319-49265-0 (eBook)
DOI 10.1007/978-3-319-49265-0
LibraryofCongressControlNumber:2016957500
MathematicsSubjectClassification(2010): 30G35,30B10,30D15
©TheAuthor(s)2016
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Contents
1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... 1
2 Slice Regular Functions: Algebra.... .... .... .... .... ..... .... 7
2.1 Definition and Main Results. .... .... .... .... .... ..... .... 7
2.2 The Representation Formula. .... .... .... .... .... ..... .... 11
2.3 Multiplication of Slice Regular Functions... .... .... ..... .... 17
2.4 Quaternionic Intrinsic Functions.. .... .... .... .... ..... .... 22
2.5 Composition of Power Series.... .... .... .... .... ..... .... 24
3 Slice Regular Functions: Analysis ... .... .... .... .... ..... .... 31
3.1 Some Integral Formulas .... .... .... .... .... .... ..... .... 31
3.2 Riemann Mapping Theorem. .... .... .... .... .... ..... .... 38
3.3 Zeros of Slice Regular Functions. .... .... .... .... ..... .... 40
3.4 Modulus of a Slice Regular Function
and Ehrenpreis–Malgrange Lemma.... .... .... .... ..... .... 45
3.5 Cartan Theorem . .... ..... .... .... .... .... .... ..... .... 52
4 Slice Regular Infinite Products.. .... .... .... .... .... ..... .... 55
4.1 Infinite Products of Quaternions.. .... .... .... .... ..... .... 55
4.2 Infinite Products of Functions.... .... .... .... .... ..... .... 59
4.3 Weierstrass Factorization Theorem.... .... .... .... ..... .... 64
4.4 Blaschke Products.... ..... .... .... .... .... .... ..... .... 73
5 Growth of Entire Slice Regular Functions. .... .... .... ..... .... 77
5.1 Growth Scale ... .... ..... .... .... .... .... .... ..... .... 77
5.2 Jensen Theorem . .... ..... .... .... .... .... .... ..... .... 84
5.3 Carathéodory Theorem..... .... .... .... .... .... ..... .... 89
5.4 Growth of the H-Product of Entire Slice Regular Functions.. .... 98
5.5 Almost Universal Entire Functions.... .... .... .... ..... .... 101
5.6 Entire Slice Regular Functions of Exponential Type .. ..... .... 106
References.... .... .... .... ..... .... .... .... .... .... ..... .... 109
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 117
v
Chapter 1
Introduction
Thetheoryofholomorphicfunctionsinonecomplexvariableassumesaparticular
flavorwhenoneconsidersfunctionsthatareholomorphicontheentireplane,namely
entirefunctions.Thereasonsfortherichnessofresultsthatarepossibleinthatcontext
aremany,butcertainlyincludethefactthatforentirefunctionswecanstudygrowth
phenomenainacleanerwaythanwhatwouldbepossibleifoneweretoconsiderthe
issuesintroducedbytheboundaryofthedomainofholomorphyassociatedwithan
individualfunction.Whilethebeautifulissuesconnectedwithanalyticcontinuation
donotariseinthecontextofentirefunctions,thetheoryacquiresmuchstrengthfrom
the ability to connect the growth of the functions to the coefficients that appear in
theirTaylorseries.
The study of entire functions, in addition, has great relevance for the study of
convolutionequations,wheredifferentspacesofentirefunctionsarisenaturallyfrom
the application of the Paley Wiener theorem (in its various forms), and from the
topologicalvectorspaceapproachthatwassoinstrumentalintheworkofSchwarz,
Malgrange,Ehrenpreis,Palamodov,andHörmander.
Furtherargumentsinsupportofthestudyofentirefunctionsaresuppliedbythe
considerationthatnotonlythemostimportantelementaryfunctionsareentire(poly-
nomials,exponentials,trigonometricandhyperbolicfunctions),butalsothatmany
ofthegreatspecialfunctionsofanalysis(theJacobithetafunction,theWeierstrass
sigmafunction,theWeierstrassdoublyperiodicfunctions,amongothers)areentire.
Finally,oneknowsthatthesolutionsoflineardifferentialequationswithpolynomial
coefficients and with constant highest derivative coefficient are entire as well (this
includestheelementaryfunctions,aswellastheAiryfunctions,forexample).
While many texts exist, which are devoted exclusively to the study of entire
functions(seeforexample[34,126,134]),oneofthebestwaystounderstandthe
importanceandthebeautyofentirefunctionsisthearticlebydeBranges[82].
Onemightthereforeaskwhetherananalogousanalysiscanbedoneforfunctions
definedonthespaceHofquaternions.Asitiswellknown,thereareseveraldifferent
definitionsofholomorphicity(orregularity,asitisoftenreferredto)forfunctions
definedonquaternions.Themostwell-knowndefinitionisprobablytheonedueto
Fueter,whoexpandedontheworkofMoisil,whodefinedregularfunctionsasthose
that satisfy a first-order system of linear differential equations that generalizes the
©TheAuthor(s)2016 1
F.Colomboetal.,EntireSliceRegularFunctions,
SpringerBriefsinMathematics,DOI10.1007/978-3-319-49265-0_1
2 1 Introduction
one of Cauchy–Riemann. These functions are often referred to as Fueter regular,
and their theory is well developed, see e.g. [69, 127]. It is therefore possible to
consider those functions which are Fueter regular on the entire quaternionic space
H,andseewhetherthereareimportantpropertiesthatcanbededuced.However,the
mathematician interested in this generalization would immediately encounter the
problemthatthesefunctionsdonotadmitanaturalpowerseriesexpansion.While
thiscommentwouldneedtobebetterqualified,itisthebasicreasonwhymostof
what is known for holomorphic entire functions cannot be extended to the case of
Fueterregularentirefunctions.
In the past ten years, however, many mathematicians have devoted significant
attentiontoadifferentnotionofregularity,knownassliceregularityorslicehyper-
holomorphy.Thistheorybeganasatheoryoffunctionsfromthespaceofquaternions
H to itself whose restriction to any complex plane contained in H was in the ker-
nelofthecorrespondingCauchy–Riemannoperator,see[110,111].Itimmediately
appeared that this class of functions, at least the ones defined on balls centered at
theorigin,coincideswiththeclassofpolynomialsandconvergentpowerseriesof
thequaternionicvariable,previouslyintroducedin[83].Inparticular,therearevery
natural generalizations of the exponential and trigonometric functions that happen
to be entire slice regular functions. Further studies showed that on suitable open
setscalledaxiallysymmetricslicedomainsthisclassoffunctionscoincideswiththe
class of functions of the form f(q)= f(x +Iy)=α(x,y)+Iβ(x,y) when the
quaternionq iswrittenintheform x +Iy (I beingasuitablequaternionsuchthat
I2 =−1)andthepair(α,β)satisfiestheCauchy–Riemannsystemandthecondi-
tionsα(x,−y)=α(x,y),β(x,−y)=−β(x,y).Thisclassoffunctionswhenαand
β arereal,quaternionicor,moreingeneral,Cliffordalgebravaluediswellknown:
theyaretheso-calledholomorphicfunctionsofaparavectorvariable,see[127,150],
whichwerelaterstudiedinthesettingofrealalternativealgebrasin[117].
Buttherearedeeperreasonswhythesefunctionsarearelevantsubjectofstudy.
Maybe the most important point is to notice that slice regular functions and their
Cliffordvaluedcompanions,theslicemonogenicfunctions(see[72,73]),havesur-
prising applications in operator theory. In particular, one can use these functions
(collectivelyreferredtoasslicehyperholomorphicfunctions)todefineacompletely
newandverypowerfulslicehyperholomorphicfunctionalcalculus(whichextends
theRiesz–Dunfordfunctionalcalculustoquaternionicoperatorsandton-tuplesof
noncommutingoperators).Thesepossibleapplicationsinoperatortheoryhavegiven
greatimpulsetothetheoryoffunctions.
In particular, the Cauchy formulas with slice hyperholomorphic kernels are the
basictoolsneededtoextendtheRiesz–Dunfordfunctionalcalculustoquaternionic
operatorsandton-tuplesofnotnecessarilycommutingoperators.Theyalsoleadto
thenotionofS-spectrumwhichturnedouttobethecorrectnotionofspectrumforall
applications in different areas of quaternionic functional analysis and in particular
tothequaternionicspectraltheorem.
The function theory of sliceregular and slicemonogenic functions was developed
inanumberofpapers,seethelistofreferences(notethatsomeofthereferencesare
notexplicitlyquotedinthetext)andthecommentsbelow.Ithasalsobeenextended
1 Introduction 3
to vector-valued and more in general operator-valued functions, see [23] and the
referencestherein.Themonographs[23,76,109]arethemainsourcesforthestudy
ofslicehyperholomorphicfunctionsandtheirapplications.
It should be pointed out that the theory of slice hyperholomorphic functions is
differentfromthemoreclassicaltheoryoffunctionsinthekerneloftheDiracoperator
[35,69,84],theso-calledmonogenicfunctions.Whilethelatterisarefinementof
harmonic analysis in several variables, the former has many applications among
whichoneofthemostimportantisinquaternionicquantummechanics,see[3,32,
89, 91, 128]. Slice hyperholomorphic functions and monogenic functions can be
relatedusingtheFuetertheorem,see[51,56,64,86]andtheRadontransform,see
[149]andthemorerecent[36].
Togiveaflavoroftheseveralresultsavailableintheliteratureinthecontextofslice
regular functions, with no claim of completeness, we gather the references in the
variousareasofresearch.
Function theory. The theory of slice regular functions was developed in [47, 105,
107, 108, 110, 111, 113, 114], in particular, their zeros were treated in [104, 106,
112]whilefurtherpropertiescanbefoundin[33,44,85,93,94,103,141,142,151,
152,153,155].SlicemonogenicfunctionswithvaluesinaCliffordalgebraandtheir
mainpropertieswerestudiedin[58,72,73,74,75,78,81,123,156].Approximation
ofslicehyperholomorphicfunctionsiscollectedin[77,95,96,97,98,99,100,101,
146].Thecaseofseveralvariableswastreatedin[2,79],butalotofworkhasstill
tobedoneinthisdirectionsincethetheoryisatthebeginning.Thegeneralization
ofsliceregularitytorealalternativealgebraswasdevelopedin[26,117,118,119,
120] and, finally, some results related to the global operator associated with slice
hyperholomorphicityarein[53,70,80,121].
Function spaces. Several function spaces have been studied in this framework. In
particular,thequaternionicHardyspaces H2(Ω),whereΩ isthequaternionicunit
ballBorthehalfspaceH+ ofquaternionswithpositiverealpart,togetherwiththe
Blaschkeproductsarein[14,15,16]andfurtherpropertiesarein[28,29,30].The
Hardyspaces Hp(B), p >2,areconsideredin[145].TheBergmanspacescanbe
foundin[50,52,54]andtheFockspacein[24].WeightedBergmanspaces,Bloch,
Besov,andDirichletspacesontheunitballBareconsideredin[38].Innerproduct
spacesandKreinspacesinthequaternionicsettingarestudiedin[21].
Groupsandsemigroupsofoperators.Thetheoryofgroupsandsemigroupsofquater-
nionicoperatorshasbeendevelopedandstudiedin[22,62,125].
Functionalcalculi.Thereexistsatleastfivefunctionalcalculiassociatedwithslice
hyperholomorphicity.Foreachonewehaveaquaternionicversionandaversionfor
n-tuplesofoperators.The S-functionalcalculus,see[9,45,48,49,59,60,61,71],
is the analog of the Riesz–Dunford functional calculus in the quaternionic setting.
Further developments are in [40, 42]. The SC-functional calculus, see [65], is the
commutative version of the S-functional calculus. For the functional calculus for
groupsofquaternionicoperatorsbasedontheLaplace–Stieltjestransform,see[8].
The H∞ functional calculus based on the S-spectrum, see [25], is the analog, in
thissetting,ofthecalculusintroducedbyA.McIntosh,see[138].TheF-functional
calculus,see[18,41,63,67,68],whichisbasedontheFuetermappingtheoremin
4 1 Introduction
integralform,isamonogenicfunctionalcalculusinthespiritoftheonedevelopedin
[129,130,131,135,139],butitisassociatedwithslicehyperholomorphicity.Finally,
the W-functional calculus, see [57], is a monogenic plane wave calculus based on
slicehyperholomorphicfunctions.
Spectraltheory.ThespectraltheorembasedontheS-spectrumforboundedandfor
unboundedquaternionicnormaloperatorsonaHilbertspacewasdevelopedin[10,
11,116].Thecaseofquaternionicnormalmatriceswasprovedin[90]anditisbased
on the right spectrum. Note that the right spectrum is equal to the S-spectrum in
thefinitedimensionalcase.Thecontinuousslicefunctionalcalculusinquaternionic
Hilbertspacesisstudiedin[115].
SchurAnalysis.Thisisaverywidefieldthathasbeendevelopedinthelastfiveyears
in the slice hyperholomorphic setting. Schur analysis originates with the works of
Schur, Herglotz, and others and can be seen as a collection of topics pertaining to
Schur functions and their generalizations; for a quick introduction in the classical
caseseeforexample[5].Fortheslicehyperholomorphiccasesee[1,2,6,7,12,13,
14,15,16,17,19,20]andalsotheforthcomingmonograph[23].
Sincetheliteratureinthefieldissovast,itisnaturaltoaskwhetherthedeeptheoryof
holomorphicentirefunctionscanbereconstructedforsliceregularfunctions.Asthis
bookwilldemonstrate,theanswerispositive,andourcontributionhereconsistsin
showingthewaytoacompletetheoryofentiresliceregularfunctions.Itisprobably
safe to assert that this monograph is only the first step, and in fact we only have
chosenafairlylimitedsubsetofthegeneraltheoryofholomorphicentirefunctions,
todemonstratethefeasibilityofourproject.Weexpecttoreturntootherimportant
topicsinasubsequentvolume.
Thismonographcontainsfourchapters,besidesthisintroduction.InChapters2
and3weintroduceanddiscussthealgebraandtheanalysisofsliceregularfunctions.
Whilemostoftheresultsinthosechaptersarewellknown,andcanbefoundinthe
literature, see e.g. [23, 76, 109], we repeated them to make the monograph self-
contained.However,thereareafewnewobservations(e.g.,inSection2.5and3.2
where we tackle the composition of slice regular functions and also the Riemann
mappingtheorem)thatdonotappearintheaforementionedmonographs.Thereare
alsoafewnewresults(forexamplewecompletethediscussiononlowerboundsfor
sliceregularfunctionsinitiatedwiththeEhrenpreis–MalgrangelemmainSection3.4,
byaddingabrand-newCartan-typetheoreminSection3.5).
Chapter4dealswithinfiniteproductsofsliceregularfunctions.Theresultsinthis
chapterareknown,butatleasttheWeierstrasstheoremreceiveshereatreatmentthat
isdifferentfromtheoneoriginallygivenin[114],seealso[109].Thistreatmentleads
alsotothedefinitionofgenusofacanonicalproduct.Thecoreofthework,however,
is Chapter5, where we study the growth of entire slice regular functions, and we
showhowsuchgrowthisrelatedtothecoefficientsofthepowerseriesexpansions
that these functions have. This chapter contains new results, the only exception is
Section5.5 which is taken from [100]. It should be noted that the proofs we offer
arenotsimplereconstructions(ortranslations)oftheholomorphiccase.Indeed,the
noncommutativesettingcreatesaseriesofnontrivialproblemsthatforceustodefine
compositionandmultiplicationinwaysthatarenotconducivetoasimplerepetition
1 Introduction 5
of the complex case. Also, the counting of the zeros is not trivial because of the
presenceofsphericalzeroswhichhaveinfinitecardinality.
Webelievethatmuchworkstillneedstobedoneinthisdirection,andwehope
that our monograph will inspire others to turn their attention to this nascent, and
alreadysorich,newfieldofnoncommutativeanalysis.
Acknowledgements ThefirsttwoauthorsaregratefultoChapmanUniversityforthekindhospi-
talityandthesupportduringtheperiodsinwhichthisworkhasbeenwritten.Thesecondauthor
thankstheprojectFIRB2012“GeometriaDifferenzialeeTeoriaGeometricadelleFunzioni”for
partiallysupportingthisresearch.
