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Xiaoman Chen Kunyu Guo Analytic Hilbert Modules CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. © 2003 by Chapman & Hall/CRC C3995 disclaimer Page 1 Thursday, February 20, 2003 1:55 PM 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 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2003 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-399-5 Library of Congress Card Number 2003040911 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper 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,operator 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 operator 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 analytic 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 submodules 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 ©2003 CRC Press LLC 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 ©2003 CRC Press LLC 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 ©2003 CRC Press LLC