Analytic Hilbert modules

200 Pages·2003·1.512 MB·English

by  Xiaoman Chen, Kunyu Guo

Xiaoman Chen
Kunyu Guo
Analytic Hilbert Modules
CHAPMAN & HALL/CRC
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© 2003 by Chapman & Hall/CRC
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Library of Congress Cataloging-in-Publication Data
Chen, Xiaoman.
Analytic Hilbert modules / by Xiaoman Chen, Kunyu Guo.
p. cm. — (Research notes in mathematics series)
Includes bibliographical references and index.
ISBN 1-58488-399-5 (alk. paper)
1. Hilbert modules. I. Guo, Kunyu. II. Title. III. Chapman & Hall/CRC research notes
in mathematics series.
QA326.C44 2003
512¢.55—dc21 2003040911

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Contents
1Introduction
2Characteristicspacesandalgebraicreduction
2.1Characteristicspacesforidealsofpolynomials
2.2AlgebraicreductionforanalyticHilbertmodules
2.3Submoduleswithfiniterank
2.4AF-cosubmodules
2.5FinitecodimensionalsubmodulesofBergmanmodules
2.6RemarksonChapter2
3RigidityforanalyticHilbertmodules
3.1RigidityforanalyticHilbertmodules
3.2Rigidityforquotientmodules
3.3RigidityforHardysubmodulesonthepolydisk
3.4RigidityforBergmanmodules
3.5RemarksonChapter3
4EquivalenceofHardysubmodules
4.1Preliminaries
4.2UnitaryequivalenceofHardysubmodulesontheunitpolydisk
4.3SimilarityofHardysubmodulesontheunitpolydisk
4.4EquivalenceofHardysubmodulesontheunitball
4.5RemarksonChapter4
5Reproducingfunctionspacesonthecomplex n-space
5.1Algebraicreductionforquasi-invariantsubspaces
5.2SomebasicpropertiesofreproducingHilbertspacesonthe
complexplane
5.3Equivalenceofquasi-invariantsubspacesonthecomplexplane
5.4Similarityofquasi-invariantsubspacesonthecomplex n-space
5.5Quasi-invariantsubspacesoftheFockspace
5.6Finitecodimensionalquasi-invariantsubspaces
5.7Beurling’sphenomenonontheFockspace
5.8RemarksonChapter5
© 2003 by Chapman & Hall/CRC
v
6TheArvesonmodule
6.1SomebasicresultsontheArvesonmodule
6.2TheToeplitzalgebraontheArvesonspace
6.3SubmodulesoftheArvesonmodule
6.4RigidityforArvesonsubmodules
6.5RemarksonChapter6
7ExtensionsofHilbertmodules
7.1Thebasictheoryofextensions
7.2Extensionsofhypo- Sˇilovmodulesoverunitmodulusalgebras
7.3ExtensionsofHilbertmodulesandHankeloperators
7.4ExtensionsofnormalHilbertmoduleovertheballalgebra
7.5RemarksonChapter7
References
© 2003 by Chapman & Hall/CRC
Preface
Thesenotesdeal withthe aspects ofanalyticHilbert modules inseveralcom-
plex variables that have been the focus of considerable attention from the
authorsinrecentyears. This volumehasbeeninspired primarilybythework
initiated on the topic by Douglas and Paulsen in Hilbert Modules over Func-
tionAlgebras [DP]. Briefly,theHilbertmoduletheoryprovidesanappropriate
frameworkforthemulti-variableoperatortheory. Inthismodulecontext,op-
erator theory has proved mutually enriching for other areas of mathematics,
including algebra, geometry, homology theory and complex analysis in one
and several variables.
Without the help and encouragement of many friends this volume would
neverhavebeenfinished. First,wewouldliketoexpressourthankstoProfes-
sors R. G. Douglas, S. Z. Yan and S. H. Sun for their encouragement, sugges-
tionsandsupport. WearealsogratefultoProfessorJ.X.Hong,Y.N.Zhang,
Y.S.TongandJ.M.Yongforinterestandsupport. Inparticular,itisaplea-
suretothankProfessorsG.L.YuandD.C.ZhengforinvitingustoVanderbilt
University in the summer of 2001 as visiting Professors in the Mathematical
Department. Severalothershavemadecommentsandhelpfulsuggestionsand
this is a good opportunity to publicly thank them: D. Sarason, D. X. Xia,
R. R. Rochberg, J. McCarthy, B. R. Li, L. M. Ge, G. H. Gong, R. W. Yang,
K. H. Zhu, K. Izuchi and S. Z. Hou.
Wearedeeplyindebtedtotherefereesfornumeroussuggestionsthathelped
make this book more readable. We also wish to express our gratitude to
Jasmin Naim and Naomi Lynch at Chapman & Hall/CRC who helped in the
preparation of the book.
The research for this work is partially supported by the NNSFC in China,
the Shuguang Project in Shanghai, the Teaching and Research Award Pro-
gram for Outstanding Young Teachers in Higher Education Institutions of
MOE, P. R. China and the Laboratory of Mathematics for Nonlinear Science
at Fudan University.
KunyuGuo: DepartmentofMathematics,FudanUniversity,Shang-
hai, 200433, P. R. China, E-mail: [email protected]
XiaomanChen: InstituteofMathematics,FudanUniversity,Shang-
hai, 200433, P. R. China, E-mail: [email protected]
© 2003 by Chapman & Hall/CRC
vii
Chapter 1
Introduction
These notes arose out of a series of research papers completed by the authors
and others. This volume is devoted to recent developments in some topics of
analytic Hilbert modules.
In the last decades, the study of nonself-adjoint operator algebras has en-
joyedconsiderablesuccess, butthedevelopmenthaslargelyexcludedspectral
theory. Multi-variable spectral theory can be viewed by analogy as “non-
commutative algebra geometry,” and such a development was the goal of the
module approach to the multi-variable operator theory presented in Hilbert
Modules over Function Algebras by Douglas and Paulsen [DP].
The theory of Hilbert modules over function algebras, largely due to
R. Douglas, provides an appropriate framework to the multi-variable oper-
ator theory. In this module context, function algebras play important roles
rather than “n-tuples,” and one considers Hilbert spaces as modules over it.
Thechangeofviewpointhassomeremarkableconsequences. Attheveryleast,
itfacilitatestheintroductionoftechniquesandmethodsdrawnfromalgebraic
geometry, homology theory and complex analysis in one variable and several
variables, etc.
Now we will simply describe the background and developments of ana-
lytic Hilbert modules; of course, the description is never complete. Let T be
the unit circle in the complex plane and let L2(T) be the Hilbert space of
square integrable functions, with respect to arc-length measure. Recall that
the Hardy space H2(D) over the open unit disk D is the closed subspace of
L2(T) spanned by the nonnegative powers of the coordinate function z. In
thelanguageoftheHilbertmodule[DP], theHardyspaceH2(D)isamodule
over the disk algebra A(D) with multiplication defined pointwise on T. If M
isasubmoduleofH2(D)(whichmeansthatM isinvariantforthemultiplica-
tion by polynomials), then Beurling’s theorem [Beu] says that there exists an
inner function η such that M = ηH2(D). Therefore, each submodule M of
the Hardy module H2(D) has the form M =ηH2(D). For the Hardy module
H2(Dn) in the multi-variable version, a natural problem is to consider the
structureofsubmodules. However, onequicklyseesthataBeurling-likechar-
acterization is impossible [Ru1]. Perhaps the earliest results showing that a
straightforward generalization of Beurling’s theorem fails in H2(Dn) involve
considering cyclic vectors and are due to Ahern and Clark [AC].
©2003 CRC Press LLC
Theorem 1.0.1 (Ahern-Clark) Iff ,...,f areinH2(Dn)withk <n,then
1 k
the submodule generated by f ,...,f is either all of H2(Dn) or is of infinite
1 k
codimension.
Therefore, in particular, no submodule of finite codimension in H2(Dn) can
be of the form ηH2(Dn). However, there are infinitely many submodules of
finite codimension. To describe these we let C denote the ring of complex
polynomials in several variables. For an ideal I in C, we set Z(I)={z ∈Cn :
p(z)=0, ∀p∈I}, the zero variety of I. We also let [I] be the closure of I in
H2(Dn).
Theorem 1.0.2 (Ahern-Clark) Suppose M is a submodule of H2(Dn) of
codimension k <∞. Then C∩M is an ideal in the ring C such that
(1) C∩M is dense in M;
(2) dim C/C∩M =k;
(3) Z(C∩M) is finite and lies in Dn.
Conversely,ifI isanidealoftheringC withZ(I)beingfinite,andZ(I)⊂Dn,
then [I] is a submodule of H2(Dn) with the same codimension as I in C and
[I]∩C =I.
The above theorem shows that the description of finite codimensional sub-
modules is, at least conceptually, an algebraic matter. We also get our first
glimpse of the interplay between the algebra and analysis. “Algebraic reduc-
tion,” the starting point of this topic, is Theorem 1.0.2. Along these lines,
many authors, including Douglas, Paulsen, Sah and Yan [DP, DPSY], Axler
and Bourdon [AB], Bercovici, Foias and Pearcy [BFP], Agrawal and Salinas
[AS], Guo [Guo1, Guo2, Guo4, Guo12], Putinar and Salinas [PS], Putinar
[Pu1, Pu2], and Chen and Douglas [CD1, CD2], have developed a series of
techniquestoinvestigatealgebraicreductionforanalyticHilbertmodulesover
some domains.
ForHilbertmodulesinseveralvariables,mostofthemarisefromreproduc-
inganalyticfunctionspaces. Thisleadsustointroducethenotionofanalytic
Hilbert modules. Let Ω be a bounded nonempty open subset of Cn, Hol(Ω)
denote the ring of analytic functions on Ω, and X be a Banach space con-
tainedinHol(Ω). WecallX tobeareproducingΩ-spaceifX contains1and
if for each λ ∈ Ω the evaluation functional, E (f) = f(λ), is a continuous
λ
linear functional on X. We say X is a reproducing C-module on Ω if X is a
reproducing Ω-space, and for each p ∈ C and each x ∈ X, px is in X. Note
that, by a simple application of the closed graph theorem, the operator T
p
defined to be multiplication by p is bounded on X. Note also that C ⊂ X
follows from the fact that 1 is in X. For λ∈Cn, one says that λ is a virtual
point of X provided that the homomorphism p (cid:55)→ p(λ) defined on C extends
toaboundedlinearfunctionalonX. SinceX isareproducingΩ-space,every
©2003 CRC Press LLC
pointinΩisavirtualpoint.Weuse vp(X)todenotethecollectionofvirtual
points,then vp(X)⊇Ω.Finallywesaythat X isananalyticHilbertmodule
on Ωifthefollowingconditionsaresatisfied:
(1) X isareproducing C-moduleonΩ;
(2) C isdensein X;
(3) vp(X)=Ω.
InChapter2,wegivearatherextendedconsiderationtoalgebraicre-
ductionofanalyticHilbertmodulesbyusingthecharacteristicspacethe-
ory.Thecharacteristicspacetheoryinseveralvariables,developedbyGuo
[Guo1,Guo2,Guo4,Guo12],isanaturalgeneralizationofthefundamental
theoremofalgebra.Thistheoryallowsustotreatalgebraicreductionof
analyticHilbertmodulesinseveralvariables(seeTheorem2.2.5,Corollary
2.2.6)uniformly.Infact,forthetopicsdiscussedinChapters2 through5,
ourmaintechniqueisthecharacteristicspacetheory.Let X beananalytic
Hilbertmoduleover adomainΩ.Recallthat asubmodule M of X isan
AF-cosubmoduleif M isequaltotheintersectionofallfinitecodimensional
submodulesthatcontain M.Wefoundthat AF-cosubmodulesenjoymany
propertiesofsubmodulesoffinitecodimension.Therefore,analgebraicre-
ductiontheoremfor AF-cosubmoduleswaspro ved(seeTheorem2.4.2).
Itiswellknownthat“rank”isoneoftheimportantinvariantsofHilbert
modules.Submoduleswithfiniterankenjoymanyalgebraicproperties,for
example,theHilbert-SamuelpolynomialsintroducedbyDouglasandYan
[DY2].Inaddition,wepro vedthatif M isasubmoduleoftheanalytic
Hilbertmodule X overthedomainΩgenerated by f ,...,f ,thenforeach
1 k (cid:80)
f ∈ M,thereareanalyticfunctions g ,...g onΩsuchthat f = k f g
1 k i=1 i i
(seeTheorem2.3.3).ThisresultwillbeappliedtostudyequivalenceofHardy
submodulesoverthepolydiskinChapter4.
Chapter 3exploresthetopicof“rigidity”ofanalyticHilbertmodulesin
several variables. By rigidity we mean that an analytic Hilbert module in
several variables exhibits a much more rigid structure than is expected from
the single-variable theory. This section starts by introducing the following
definitions.
Definition. Let M , M be two submodules of X on Ω. We say that
1 2
1. they are unitarily equivalent if there exists a unitary module map X :
M → M , that is, X is a unitary operator and for any polynomial p,
1 2
X(ph)=pX(h), ∀h∈M ;
1
2. theyaresimilarifthereexistsaninvertiblemodulemapX : M →M ;
1 2
3. they are quasi-similar if there exist module maps X : M → M and
1 2
Y : M →M with dense ranges.
2 1
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ByBeurling’stheorem,submodulesof H2(D)areallunitarilyequivalent.
However,forthehigherdimensionalHardymodule H2(Dn),anearlierresult
onnon-unitarilyequivalentsubmodules,duetoBerger,CoburnandLebow
[BCL],startedtoindicatejusthowmuchrigiditythereisamongthosesub-
moduleswithfinitecodimension.Theyconsideredtherestrictionofmultipli-
cationbythecoordinatefunctionstosubmodulesobtainedastheclosureof
certainidealsofpolynomialsinseveralvariableshavingtheoriginaszeroset.
Byapplyingtheirresultsoncommutingisometries,theyshowedthatdiffer-
entidealsyieldinequivalentsubmodules.AlmostatthesametimeHastings
[Ha2]evenshowedthat[ z−w]isneverquasi-similarto H2(D2),where[ z−w]
denotesthesubmodulegeneratedby z −w.Therigiditytheoremforfinite
codimensionalsubmodulesof H2(Dn)isduetoAgrawal,ClarkandDouglas
[ACD].Thistheoremshowedthattwosubmodulesoffinitecodimensionare
unitarilyequivalentonlyiftheyareequal.Byusinglocalizationtechniques,
Douglas,Paulsen,SahandYan[DPSY]gaveacompletegeneralizationof
therigiditytheoremabove.Actuallytheyobtained astrongerresult[DPSY,
Corollary2.8]thanstatedhere.
Theorem1.0.3(DPSY) Let X beananalyticHilbertmoduleoverthedo-
main Ω,andlet I and I beidealsofthepolynomialsring C in n-complex
1 2
variables.Supposethat dim Z(I ) ≤ n−2,andeachalgebraiccomponentof
i
Z(I ) intersects Ω for i =1 ,2. Then [I ] and [I ] arequasi-similaronlyif
i 1 2
I =I .
1 2
Theconditiondim Z(I)≤n−2issharpbecausethereexistsaunitaryequiva-
lencebetweenthesubmodules[ z]and[ w]of H2(D2)buttheyareneverequal.
Notethatthisconditionisalsoequivalenttothefactthattheideal I hasthe
greatestcommondivisor1(seeChapter3).Therefore,Theorem1.0.3pro-
vides more examples for nonquasi-similar submodules. A simple example is
that the submodules [z ,z ] and [z ,z ] of the Hardy module H2(D3) are not
1 2 1 3
quasi-similar. By using the characteristic space theory, Guo [Guo1] obtained
a generalization for the rigidity theorem above (see Theorem 3.1.6).
Anotherkindofrigiditytheoremcloselyrelatedtotheprecedingconsidera-
tion (but different, since the submodules involved do not possess an algebraic
model) was obtained by Douglas and Yan [DY1]. Given a submodule M of
H2(Dn), let Z(M)={z ∈Dn : f(z)=0, ∀f ∈M}. For f in H2(Dn), let σ
f
betheuniquesingularmeasureonn-torusTn forwhichP (log|f|−σ )isthe
z f
least harmonic majorant of log|f|, where P denotes the Poisson integral.
z
Theorem 1.0.4 (Douglas-Yan) Let M and M be submodules of H2(Dn)
1 2
such that
(1) the Hausdorff dimension of Z(M ) is at most n−2, i=1,2;
i
(2) inf{σ : f ∈M }=inf{σ : f ∈M }.
f 1 f 2
Then M and M are quasi-similar if and only if M =M .
1 2 1 2
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Theabovetheoremreliesheavilyonthefunctiontheoryof H2(Dn),andal-
thoughsomegeneralizationsareknown[Guo8,orseeSection3.3],itisnot
clearyetwhat ageneralizationofthisresultistootheranalyticHilbertmod-
ules.
Chapter3isdevotedtotheseremarkablerigidityfeaturesofanalyticHilbert
modulesinseveralvariables.Fromananalyticpointofview,appearanceof
thisfeatureisnaturalbecauseoftheHartogsphenomenoninseveralvariables.
Fromanalgebraicpointofview,thereasonmaybethatsubmodulesarenot
singlygenerated.
Theequivalenceproblemis,ingeneral,quitedifficult.Ononehand,under
thehypothesesofTheorem1.0.3,theequivalencenecessarilypreservesthe
ideal.Ontheotherhand,forthesubmodules[ z]and[ w]of H2(D2)the
unitaryequivalence U :[ z]→[w]definedbymultiplication¯ zw doesnoteven
preservethezerosetsoftheideals.Yanhasexaminedthecaseofhomogeneous
principalideals[Yan1].
Theorem1.0.5(Yan) Let p; q betwohomogeneouspolynomials.Then [p]
and [q],asthesubmodulesof H2(Dn),areunitarilyequivalentifandonlyif
thereexists aconstant c suchthat |p|=c|q| on Tn;similarifandonlyifthe
quotient |p|/|q| isboundedaboveandbelowon Tn.
Whileinthecaseoftheunitball Bn,ChenandDouglas[CD1]provedthat
[p]and[ q]arequasi-similaronlyif p = cq forsomeconstant c.Fromthese
factsonefindsthattheequivalenceproblemdependsheavilyonthegeometric
propertiesofdomains.
Chapter4considerstheequivalenceproblemofHardysubmodulesgener-
atedbypolynomialsinthecasesofboththepolydiskandtheunitball. LetI
be an ideal of the polynomials ring C. Since C is a Noetherian ring [AM, ZS],
theidealI isgeneratedbyfinitelymanypolynomials. ThisimpliesthatI has
a greatest common divisor p. Thus, I can be uniquely written as I = pL,
which is called the Beurling form of I. The following classification theorems
were proved in [Guo2](see Theorem 4.2.1, Corollary 4.2.2 and Theorem 4.4.2
in this volume).
Theorem 1.0.6 (Guo) Let I and I be two ideals of polynomials and let
1 2
I =p L , I =p L be their Beurling forms. Then
1 1 1 2 2 2
(1) onthepolydiskDn, [I ]and[I ], assubmodulesofH2(Dn), areunitarily
1 2
equivalent if and only if there exist two polynomials q , q with Z(q )∩
1 2 1
Dn = Z(q )∩Dn = ∅ such that |p q | = |p q | on Tn, and [p L ] =
2 1 1 2 2 1 1
[p L ].
1 2
(2) ontheunitballB , [I ]and[I ], assubmodulesofH2(B ), areunitarily
n 1 2 n
equivalent only if [I ]=[I ].
1 2
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