Operator Theory: Advances andApplicationsVol. 157

Editor:I. Gohberg

H. G. Kaper (Argonne)S. T. Kuroda (Tokyo)P. Lancaster (Calgary)L. E. Lerer (Haifa)B. Mityagin (Columbus)V. V. Peller (Manhattan, Kansas)L. Rodman (Williamsburg)J. Rovnyak (Charlottesville)D. E. Sarason (Berkeley)I. M. Spitkovsky (Williamsburg)S. Treil (Providence)H. Upmeier (Marburg)S. M. Verduyn Lunel (Leiden)D. Voiculescu (Berkeley)H. Widom (Santa Cruz)D. Xia (Nashville)D. Yafaev (Rennes)

Honorary and AdvisoryEditorial Board:C. Foias (Bloomington)P. R. Halmos (Santa Clara)T. Kailath (Stanford)P. D. Lax (New York)M. S. Livsic (Beer Sheva)

Editorial Office:School of MathematicalSciencesTel Aviv UniversityRamat Aviv, Israel

Editorial Board:D. Alpay (Beer-Sheva)J. Arazy (Haifa)A. Atzmon (Tel Aviv)J. A. Ball (Blacksburg)A. Ben-Artzi (Tel Aviv)H. Bercovici (Bloomington)A. Böttcher (Chemnitz)K. Clancey (Athens, USA)L. A. Coburn (Buffalo)K. R. Davidson (Waterloo, Ontario)R. G. Douglas (College Station)A. Dijksma (Groningen)H. Dym (Rehovot)P. A. Fuhrmann (Beer Sheva)B. Gramsch (Mainz)G. Heinig (Chemnitz)J. A. Helton (La Jolla)M. A. Kaashoek (Amsterdam)

Birkhäuser VerlagBasel . Boston . Berlin

Operator Theory, Systems Theoryand Scattering Theory:Multidimensional Generalizations

Daniel AlpayVictor VinnikovEditors

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Bibliographic information published by Die Deutsche BibliothekDie Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is availablein the Internet at <http://dnb.ddb.de>.

ISBN 3-7643-7212-5 Birkhäuser Verlag, Basel – Boston – Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilmsor in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, SwitzerlandPart of Springer Science+Business MediaPrinted on acid-free paper produced from chlorine-free pulp. TCF ∞Cover design: Heinz Hiltbrunner, BaselPrinted in GermanyISBN 10: 3-7643-7212-5ISBN 13: 978-3-7643-7212-59 8 7 6 5 4 3 2 1 www.birkhauser.ch

Editors:

Daniel AlpayVictor VinnikovDepartment of MathematicsBen-Gurion University of the NegevP.O. Box 653Beer Sheva 84105Israele-mail: dany@math.bgu.ac.il

vinnikov@math.bgu.ac.il

2000 Mathematics Subject Classification 47A13, 47A40, 93B28

Contents

Editorial Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

J. Ball and V. VinnikovFunctional Models for Representations of the Cuntz Algebra . . . . . . . . . 1

T. Banks, T. Constantinescu and J.L. JohnsonRelations on Non-commutative Variables andAssociated Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

M. BessmertnyıFunctions of Several Variables in the Theory of FiniteLinear Structures. Part I: Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

S. Eidelman and Y. KrasnovOperator Methods for Solutions of PDE’sBased on Their Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

D.S. Kalyuzhnyı-VerbovetzkiıOn the Bessmertnyı Class of Homogeneous PositiveHolomorphic Functions on a Product of Matrix Halfplanes . . . . . . . . . . . 139

V. Katsnelson and D. VolokRational Solutions of the Schlesinger System andIsoprincipal Deformations of Rational Matrix Functions II . . . . . . . . . . . 165

M.E. Luna–Elizarraras and M. ShapiroPreservation of the Norms of Linear Operators Actingon some Quaternionic Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

P. Muhly and B. SolelHardy Algebras Associated with W ∗-correspondences(Point Evaluation and Schur Class Functions) . . . . . . . . . . . . . . . . . . . . . . . . 221

M. PutinarNotes on Generalized Lemniscates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

M. Reurings and L. RodmanOne-sided Tangential Interpolation for Hilbert–SchmidtOperator Functions with Symmetries on the Bidisk . . . . . . . . . . . . . . . . . . 267

F.H. SzafraniecFavard’s Theorem Modulo an Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Operator Theory:Advances and Applications, Vol. 157, vii–xvic© 2005 Birkhauser Verlag Basel/Switzerland

Editorial Introduction

Daniel Alpay and Victor Vinnikov

La seduction de certains problemes vient de leur

defaut de rigueur, comme des opinions discor-

dantes qu’ils suscitent: autant de difficultes dont

s’entiche l’amateur d’Insoluble.

(Cioran, La tentation d’exister, [29, p. 230])

This volume contains a selection of papers on various aspects of operator theoryin the multi-dimensional case. This last term includes a wide range of situationsand we review the one variable case first.

An important player in the single variable theory is a contractive analytic func-tion on the open unit disk. Such functions, often called Schur functions, have arich theory of their own, especially in connection with the classical interpolationproblems. They also have different facets arising from their appearance in differentareas, in particular as:• characteristic operator functions, in operator model theory. Pioneering works

include the works of Livsic and his collaborators [54], [55], [25], of Sz. Nagyand Foias [61] and of de Branges and Rovnyak [23], [22].• scattering functions, in scattering theory. We mention in particular the Lax–

Phillips approach (see [53]), the approach of de Branges and Rovnyak (see[22]) and the inverse scattering problem of network theory [38]; for a solutionof the latter using reproducing kernel Hilbert space methods, see [8], [9].• transfer functions, in system theory. It follows from the Bochner–Chandra-

sekharan theorem that a system is linear, time-invariant, and dissipative ifand only if it has a transfer function which is a Schur function. For moregeneral systems (even multi-dimensional ones) one can make use of Schwartz’kernel theorem (see [76], [52]) to get the characterisation of invariance undertranslation; see [83, p. 89, p. 130].

There are many quite different approaches to the study of Schur functions, theirvarious incarnations and related problems, yet it is basically true that there is onlyone underlying theory.

viii D. Alpay and V. Vinnikov

One natural extension of the single variable theory is the time varying case, whereone (roughly speaking) replaces the complex numbers by diagonal operators andthe complex variable by a shift operator; see [7], [39].

The time varying case is still essentially a one variable theory, and the variousapproaches of the standard one variable theory generalize together with their in-terrelations. On the other hand, in the multi-dimensional case there is no longera single underlying theory, but rather different theories, some of them looselyconnected and some not connected at all. In fact, depending on which facet ofthe one-dimensional case we want to generalize we are led to completely differentobjects and borderlines between the various theories are sometimes vague. Thedirections represented in this volume include:

• Interpolation and realization theory for analytic functions on the polydisk.This originates with the works of Agler [2], [1]. From the view point of sys-tem theory, one is dealing here with the conservative version of the systemsknown as the Roesser model or the Fornasini–Marchesini model in the multi-dimensional system theory literature; see [71], [46].• Function theory on the free semigroup and on the unit ball of CN . From

the view point of system theory, one considers here the realization problemfor formal power series in non-commuting variables that appeared first inthe theory of automata, see Schutzenberger [74], [75] and Fliess [44], [45](for a good survey see [17]), and more recently in robust control of linearsystems subjected to structured possibly time-varying uncertainty (see Beck,Doyle and Glover [15] and Lu, Zhou and Doyle [59]). In operator theory, twomain parallel directions may be distinguished; the first direction is along thelines of the works of Drury [43], Frazho [47], [48], Bunce [26], and especiallythe vast work of Popescu [65], [63], [64], [66], where various one-dimensionalmodels are extended to the case of several non-commuting operators. Anotherdirection is related to the representations of the Cuntz algebra and is alongthe line of the works of Davidson and Pitts (see [36] and [37]) and Bratelliand Jorgensen [24]. When one abelianizes the setting, one obtains resultson the theory of multipliers in the so-called Arveson space of the ball (see[12]), which are closely related with the theory of complete Nevanlinna–Pickkernels; see the works of Quiggin [70], McCullough and Trent [60] and Aglerand McCarthy [3]. We note also connections with the theory of wavelets andwith system theory on trees; see [16], [10].• Hyponormal operators, subnormal operators, and related topics. Though nom-

inally dealing with a single operator, the theory of hyponormal operators andof certain classes of subnormal operators has many features in common withmultivariable operator theory. We have in mind, in particular, the works ofPutinar [68], Xia [81], and Yakubovich [82]. For an excellent general survey ofthe theory of hyponormal operators, see [80]. Closely related is the principalfunction theory of Carey and Pincus, which is a far reaching developmentof the theory of Kreın’s spectral shift function; see [62], [27], [28]. Another

Editorial Introduction ix

closely related topic is the study of multi-dimensional moment problems; ofthe vast literature we mention (in addition to [68]) the works of Curto andFialkow [33], [34] and of Putinar and Vasilescu [69].• Hyperanalytic functions and applications. Left (resp. right) hyperanalytic

functions are quaternionic-valued functions in the kernel of the left (resp.right) Cauchy–Fueter operator (these are extensions to R4 of the operator∂∂x + i ∂

∂y ). The theory is non-commutative and a supplementary difficultyis that the product of two (say, left) hyperanalytic functions need not beleft hyperanalytic. Setting the real part of the quaternionic variable to bezero, one obtains a real analytic quaternionic-valued function. Conversely,the Cauchy–Kovalevskaya theorem allows to associate (at least locally) toany such function a hyperanalytic function. Identifying the quaternions withC2 one obtains an extension of the theory of functions of one complex variableto maps from (open subsets of) C2 into C2. Rather than two variables thereare now three non-commutative non-independent hyperanalytic variables andthe counterparts of the polynomials zn1

1 zn22 are now non-commutative poly-

nomials (called the Fueter polynomials) in these hyperanalytic variables. Theoriginal papers of Fueter (see, e.g., [50], [49]) are still worth a careful reading.• Holomorphic deformations of linear differential equations. One approach to

study of non-linear differential equations, originating in the papers of Schle-singer [73] and Garnier [51], is to represent the non-linear equation as thecompatibility condition for some over-determined linear differential systemand consider the corresponding families (so-called deformations) of ordinarylinear equations. From the view point of this theory, the situation when thelinear equations admit rational solutions is exceptional: the non-resonanceconditions, the importance of which can be illustrated by Bolibruch’s coun-terexample to Hilbert’s 21st problem (see [11]), are not met. However, anal-ysis of this situation in terms of the system realization theory may lead toexplicit solutions and shed some light on various resonance phenomena.

The papers in the present volume can be divided along these categories as follows:

Polydisk function theory:

The volume contains a fourth part of the translation of the unpublished thesis [18]of Bessmertnyı, which foreshadowed many subsequent developments and containsa wealth of ideas still to be explored. The other parts are available in [20], [19]and [21]. The paper of Reurings and Rodman, One-sided tangential interpolationfor Hilbert–Schmidt operator functions with symmetries on the bidisk, deals withinterpolation in the bidisk in the setting of H2 rather than of H∞.

Non-commutative function theory and operator theory:

The first paper in this category in the volume is the paper of Ball and Vinnikov,Functional models for representations of the Cuntz algebra. There, the authorsdevelop functional models and a certain theory of Fourier representation for a rep-resentation of the Cuntz algebra (i.e., a row unitary operator). Next we have the

x D. Alpay and V. Vinnikov

paper of Banks, Constantinescu and Johnson, Relations on non-commutative vari-ables and associated orthogonal polynomials, where the authors survey varioussettings where analogs of classical ideas concerning orthogonal polynomials andassociated positive kernels occur. The paper serves as a useful invitation and ori-entation for the reader to explore any particular topic more deeply. In the paperof Kalyuzhnyı-Verbovetzkiı, On the Bessmertnyı class of homogeneous positiveholomorphic functions on a product of matrix halfplanes, a recent investigationof the author on the Bessmertnyı class of operator-valued functions on the openright poly-halfplane which admit a so-called long resolvent representation (i.e., aSchur complement formula applied to a linear homogeneous pencil of operatorswith positive semidefinite operator coefficients), is generalized to a more general“non-commutative” domain, a product of matrix halfplanes. The study of the Bess-mertnyı class (as well as its generalization) is motivated by the electrical networkstheory: as shown by M.F. Bessmertnyı [18], for the case of matrix-valued func-tions for which finite-dimensional long resolvent representations exist, this classis exactly the class of characteristic functions of passive electrical 2n-poles whereimpedances of the elements of a circuit are considered as independent variables.Finally, in the paper Hardy algebras associated with W ∗-correspondences (pointevaluation and Schur class functions), Muhly and Solel deal with an extension ofthe non-commutative theory from the point of view of non-self-adjoint operatoralgebras.

Hyponormal and subnormal operators and related topics:

The paper of Putinar, Notes on generalized lemniscates, is a survey of the theoryof domains bounded by a level set of the matrix resolvent localized at a cyclicvector. The subject has its roots in the theory of hyponormal operators on the onehand and in the theory of quadrature domains on the other. While both topics arementioned in the paper, the main goal is to present the theory of these domains(that the author calls “generalized lemniscates”) as an independent subject matter,with a wealth of interesting properties and applications. The paper of Szafraniec,Orthogonality of polynomials on algebraic sets, surveys recent extensive work ofthe author and his coworkers on polynomials in several variables orthogonal on analgebraic set (or more generally with respect to a positive semidefinite functional)and three term recurrence relations. As it happens often the general approachsheds new light also on the classical one-dimensional situation.

Hyperanalytic functions:

In the paper Operator methods for solutions of differential equations based ontheir symmetries, Eidelman and Krasnov deal with construction of explicit solu-tions for some classes of partial differential equations of importance in physics, suchas evolution equations, homogeneous linear equations with constant coefficients,and analytic systems of partial differential equations. The method used involvesan explicit construction of the symmetry operators for the given partial differen-tial operator and the study of the corresponding algebraic relations; the solutions

Editorial Introduction xi

of the partial differential equation are then obtained via the action of the sym-metry operators on the “simplest” solution. This allows to obtain representationsof Clifford-analytic functions in terms of power series in operator indeterminates.Luna–Elizarraras and Shapiro in Preservation of the norms of linear operators act-ing on some quaternionic function spaces consider quaternionic analogs of someclassical real spaces and in particular compare the norms of operators in the orig-inal space and in the quaternionic extension.

Holomorphic deformations of linear differential equations:

This direction is represented in the present volume by the paper of Katsnelson andVolok, Rational solutions of the Schlesinger system and rational matrix functionsII, which presents an explicit construction of the multi-parametric holomorphicfamilies of rational matrix functions, corresponding to rational solutions of theSchlesinger non-linear system of partial differential equations.

There are many other directions that are not represented in this volume. Withoutthe pretense of even trying to be comprehensive we mention in particular:

• Model theory for commuting operator tuples subject to various higher-ordercontractivity assumptions; see [35], [67].• A multitude of results in spectral multivariable operator theory (many of

them related to the theory of analytic functions of several complex variables)stemming to a large extent from the discovery by Taylor of the notions of thejoint spectrum [78] and of the analytic functional calculus [77] for commutingoperators (see [32] for a survey of some of these).• The work of Douglas and of his collaborators based on the theory of Hilbert

modules; see [42], [40], [41].• The work of Agler, Young and their collaborators on operator theory and

realization theory related to function theory on the symmetrized bidisk, withapplications to the two-by-two spectral Nevanlinna–Pick problem; see [5], [4],[6].• Spectral analysis and the notion of the characteristic function for commuting

operators, related to overdetermined multi-dimensional systems. The mainnotion is that of an operator vessel, due to Livsic; see [56], [57], [58]. Thisturns out to be closely related to function theory on a Riemann surface; see[79],[13].• The work of Cotlar and Sadosky on multievolution scattering systems, with

applications to interpolation problems and harmonic analysis in several vari-ables; see [30], [31], [72].

Acknowledgments

This volume has its roots in a workshop entitled Operator theory, system theoryand scattering theory: multi-dimensional generalizations, 2003, which was held atthe Department of Mathematics of Ben-Gurion University of the Negev during theperiod June 30–July 3, 2003. It is a pleasure to thank all the participants for an

xii D. Alpay and V. Vinnikov

exciting scientific atmosphere and the Center of Advanced Studies in Mathemat-ics of Ben-Gurion University of the Negev for its generosity and for making theworkshop possible.

References

[1] J. Agler. Interpolation, UCSD, preprint, 1987.

[2] J. Agler. On the representation of certain holomorphic functions defined on a poly-disk. In volume 48 of Operator Theory: Advances and Applications, pp. 47–66,Birkhauser Verlag, Basel, 1990.

[3] J. Agler and J. McCarthy. Complete Nevanlinna-Pick kernels. J. Funct. Anal.,175:111–124, 2000.

[4] J. Agler, F.B. Yeh, and N.J. Young. Realization of functions into the symmetrisedbidisc. In Reproducing kernel spaces and applications, volume 143 of Oper. TheoryAdv. Appl., pp. 1–37. Birkhauser, Basel, 2003.

[5] J. Agler and N.J. Young. A commutant lifting theorem for a domain in C2 andspectral interpolation. J. Funct. Anal. 161 (1999), no. 2, 452–477.

[6] J. Agler and N.J. Young. The two-by-two spectral Nevanlinna-Pick problem. Trans.Amer. Math. Soc., 356(2):573–585 (electronic), 2004.

[7] D. Alpay, P. Dewilde, and H. Dym. Lossless inverse scattering and reproducing ker-nels for upper triangular operators. In Extension and interpolation of linear operatorsand matrix functions, pp. 61–135. Birkhauser, Basel, 1990.

[8] D. Alpay and H. Dym. Hilbert spaces of analytic functions, inverse scattering andoperator models, I. Integral Equation and Operator Theory, 7:589–641, 1984.

[9] D. Alpay and H. Dym. Hilbert spaces of analytic functions, inverse scattering andoperator models, II. Integral Equation and Operator Theory, 8:145–180, 1985.

[10] D. Alpay and D. Volok. Interpolation et espace de Hardy sur l’arbre dyadique: le casstationnaire. Comptes Rendus Mathematiques, 336:293–298, 2003.

[11] D.V. Anosov and A.A. Bolibruch. The Riemann-Hilbert problem. Aspects of Math-ematics, E22. Friedr. Vieweg & Sohn, Braunschweig, 1994.

[12] W. Arveson. Subalgebras of C∗–algebras. III. Multivariable operator theory. ActaMathematica, 181:159–228, 1998.

[13] J.A. Ball and V. Vinnikov. Overdetermined multi-dimensional systems: state spaceand frequency domain methods. In Mathematical Systems Theory in Biology, Com-munications, Computation, and Finance (Ed. J. Rosenthal and D.S. Gilliam), IMAVolumes in Mathematics and its Applications, Springer, Berlin–New York, 2003.

[14] C.L. Beck. On formal power series representations for uncertain systems. IEEETrans. Auto. Contr. 46:2 (2001), 314–319.

[15] C.L. Beck, J.C. Doyle, and K. Glover. Model reduction of multi-dimensional anduncertain systems. IEEE Trans. Auto. Contr. 41:10 (1996), 1406–1477.

[16] A. Benveniste, R. Nikoukhah, and A. Willsky. Multiscale system theory. IEEE Trans.Circuits Systems I Fund. Theory Appl., 41(1):2–15, 1994.

[17] J. Berstel and C. Reutenauer. Rational series and their languages. EATCS Mono-graphs on Theoretical Computer Science, Springer-Verlag, Berlin–New York, 1984.

Editorial Introduction xiii

[18] M.F. Bessmertnyı. Functions of several complex variables in the theory of finite linearstructures. PhD thesis, Kharkov University, Kharkov, 1982. (Russian).

[19] M.F. Bessmertnyı. On realizations of rational matrix functions of several com-plex variables. In Interpolation theory, systems theory and related topics (TelAviv/Rehovot, 1999), volume 134 of Oper. Theory Adv. Appl., pp. 157–185. Birk-hauser, Basel, 2002. Translated from the Russian by D. Alpay and V. Katsnelson.

[20] M.F. Bessmertnyı. On realizations of rational matrix functions of several complexvariables. II. In Reproducing kernel spaces and applications, volume 143 of Oper. The-ory Adv. Appl., pp. 135–146. Birkhauser, Basel, 2003. Translated from the Russianby V. Katsnelson.

[21] M.F. Bessmertnyı. On realizations of rational matrix functions of several complexvariables. III. In Current trends in operator theory and its applications (BlacksburgVA, 2002), volume 149 of Oper. Theory Adv. Appl., pp. 133–138. Birkhauser, Basel,2004. Translated from the Russian by D.S. Kalyuzhnyı-Verbovetzkiı.

[22] L. de Branges and J. Rovnyak. Canonical models in quantum scattering theory. InC. Wilcox, editor, Perturbation theory and its applications in quantum mechanics,pp. 295–392. Wiley, New York, 1966.

[23] L. de Branges and J. Rovnyak. Square summable power series. Holt, Rinehart andWinston, New York, 1966.

[24] O. Bratteli and P. Jorgensen. Wavelets through a looking glass. Applied and Numer-ical Harmonic Analysis. Birkhauser Boston Inc., Boston, MA, 2002. The world ofthe spectrum.

[25] M.S. Brodskiı and M.S. Livsic. Spectral analysis of non-self-adjoint operators andintermediate systems. Uspehi Mat. Nauk (N.S.), 13(1(79)):3–85, 1958.

[26] John W. Bunce. Models for n-tuples of noncommuting operators. J. Funct. Anal.,57(1):21–30, 1984.

[27] R.W. Carey. A unitary invariant for pairs of self-adjoint operators. J. Reine Angew.Math., 283/284:294–312, 1976.

[28] R.W. Carey and J.D. Pincus. Mosaics, principal functions, and mean motions in VonNeumann algebras. Acta Math., 138:153–218, 1977.

[29] E.M. Cioran. La tentation d’exister. Collection idees. Gallimard, 1974. First pub-lished in 1956.

[30] M. Cotlar and C. Sadosky. Generalized Bochner Theorem in algebraic scatteringsystems. In Analysis at Urbana vol.II. London Math. Soc. Lecture Notes Ser. 138(Cambridge Univ. Press, Cambridge, 1989), pp. 144–169.

[31] M. Cotlar and C. Sadosky. Integral representations of bounded Hankel forms definedin scattering systems with a multi-dimensional evolution group. Contributions to Op-erator Theory and its Applications (Mesa, AZ, 1987) (Ed. I. Gohberg, J.W. Heltonand L. Rodman), pp. 357-375, volume 35 of Operator Theory: Advances and Appli-cations, Birkhauser, Basel–Boston, 1988.

[32] R.E. Curto. Applications of several complex variables to multiparameter spectraltheory. In Surveys of some recent results in operator theory, Vol. II, pp. 25–90,Longman Sci. Tech., Harlow, 1988.

[33] R.E. Curto and L.A. Fialkow. Solution of the truncated complex moment problemfor flat data. Mem. Amer. Math. Soc. 119 (1996), no. 56.

xiv D. Alpay and V. Vinnikov

[34] R.E. Curto and L.A. Fialkow. Flat extensions of positive moment matrices: recur-sively generated relations. Mem. Amer. Math. Soc. 136 (1998), no. 648.

[35] R.E. Curto and F.-H. Vasilescu. Standard operator models in the polydisc. II. Indi-ana Univ. Math. J., 44(3):727–746, 1995.

[36] K.R. Davidson and D.R. Pitts. The algebraic structure of non-commutative analyticToeplitz algebras. Math. Ann., 311(2):275–303, 1998.

[37] K.R. Davidson and D.R. Pitts. Invariant subspaces and hyper-reflexivity for freesemigroup algebras. Proc. London Math. Soc. (3), 78(2):401–430, 1999.

[38] P. Dewilde and H. Dym. Lossless inverse scattering, digital filters, and estimationtheory. IEEE Trans. Inform. Theory, 30(4):644–662, 1984.

[39] P. Dewilde and A.-J. van der Veen. Time-varying systems and computations. KluwerAcademic Publishers, Boston, MA, 1998.

[40] R.G. Douglas. Models and resolutions for Hilbert modules. In Multivariable operatortheory (Seattle, WA, 1993), volume 185 of Contemp. Math., pp. 109–131. Amer.Math. Soc., Providence, RI, 1995.

[41] R.G. Douglas and G. Misra. Quasi-free resolutions of Hilbert modules. Integral Equa-tions Operator Theory 47 (2003), no. 4, 435–456.

[42] R.G. Douglas and V.I. Paulsen. Hilbert modules over function algebras. LongmanSci. Tech., Harlow, 1989.

[43] S.W. Drury. A generalization of von Neumann’s inequality to the complex ball. Proc.Amer. Math. Soc., 68(3):300–304, 1978.

[44] M. Fliess. Matrices de Hankel. J. Math. Pures Appl., 53:197–222, 1974 & erratum54, 1975.

[45] M. Fliess. Sur divers produits de series formelles. Bull. Soc. Math. France 102 (1974),181–191.

[46] E. Fornasini and G. Marchesini. Doubly-indexed dynamical systems: state-spacemodels and structural properties. Math. Systems Theory, 12(1):59–72, 1978/79.

[47] A.E. Frazho. Models for noncommuting operators. J. Funct. Anal., 48(1):1–11, 1982.

[48] A.E. Frazho. Complements to models for noncommuting operators. J. Funct. Anal.,59(3):445–461, 1984.

[49] R. Fueter. Analytische Funktionen einer Quaternionenvariablen. Comment. Math.Helv., 4:9–20, 1932.

[50] R. Fueter. Die Theorie der regularen Funktionen einer quaternionen Variablen. InComptes rendus du congres international des mathematiciens, Oslo 1936, Tome I,pp. 75–91, 1937.

[51] R. Garnier. Sur une classe d’equations differentielles dont les integrales generalesont leurs points critiques fixes. Compt. Rend. de l’Academie des Sciences, Paris,151:205–208, 1910.

[52] I.M. Gelfand and N.Y. Vilenkin. Generalized functions. Volume 4: Applications ofharmonic analysis. Translated from Russian. Academic Press, New York, 1964–1968.

[53] P.D. Lax and R.S. Phillips. Scattering theory (revised edition), volume 26 of Pureand Applied Mathematics. Academic Press, New York, 1989.

[54] M.S. Livsic. On the theory of isometric operators with equal deficiency indices. Dok-lady Akad. Nauk SSSR (N.S.), 58:13–15, 1947.

Editorial Introduction xv

[55] M.S. Livsic. On the theory of the elementary divisors of non-Hermitian operators.Doklady Akad. Nauk SSSR (N.S.), 60:17–20, 1948.

[56] M.S. Livsic. The inverse problem for the characteristic functions of several commut-ing operators. Integral Equations Operator Theory, 2(2):264–286, 1979.

[57] M.S. Livsic. Commuting nonselfadjoint operators and mappings of vector bundles onalgebraic curves, volume 19 of Operator Theory: Advances and Applications, pp.275–279. Birkhauser Verlag, Basel, 1986.

[58] M.S. Livsic, N. Kravitski, A. Markus, and V. Vinnikov. Commuting nonselfadjointoperators and their applications to system theory. Kluwer, 1995.

[59] W.M. Lu, K. Zhou, and J.C. Doyle. Stabilization of uncertain linear systems: AnLFT approach. IEEE Trans. Auto. Contr. 41:1 (1996), 50–65.

[60] S. McCullough and T. Trent. Invariant subspaces and Nevanlinna-Pick kernels. J.Funct. Anal., 178(1):226–249, 2000.

[61] B. Sz. Nagy and C. Foias. Harmonic analysis of operators on Hilbert spaces.Akademia Kiado, Budapest, 1966.

[62] J.D. Pincus. Some applications of operator valued analytic functions of two complexvariables. In Linear operators and approximation (Proc. Conf., Math. Res. Inst.,Oberwolfach, 1971), pp. 68–79. Internat. Ser. Numer. Math., Vol. 20. Birkhauser,Basel, 1972.

[63] G. Popescu. Characteristic functions for infinite sequences of noncommuting opera-tors. J. Operator Theory, 22(1):51–71, 1989.

[64] G. Popescu. Isometric dilations for infinite sequences of noncommuting operators.Trans. Amer. Math. Soc., 316(2):523–536, 1989.

[65] G. Popescu. Models for infinite sequences of noncommuting operators. Acta Sci.Math. (Szeged), 53(3-4):355–368, 1989.

[66] G. Popescu. Multi-analytic operators on Fock spaces. Math. Ann., 303(1):31–46,1995.

[67] S. Pott. Standard models under polynomial positivity conditions. J. Operator The-ory, 41:365–389, 1999.

[68] M. Putinar. Extremal solutions of the two-dimensional L-problem of moments. J.Funct. Anal., 136(2):331–364, 1996.

[69] M. Putinar and F.-H. Vasilescu. Solving moment problems by dimensional extension.Ann. of Math. (2) 149 (1999), no. 3, 1087–1107.

[70] P. Quiggin. For which reproducing kernel Hilbert spaces is Pick’s theorem true?Integral Equations Operator Theory, 16:244–266, 1993.

[71] R. Roesser. A discrete state-space model for linear image processing. IEEE Trans.Automatic Control, AC–20:1–10, 1975.

[72] C. Sadosky. Liftings of kernels shift-invariant in scattering systems. In HolomorphicSpaces (Ed. S. Axler, J.E. McCarthy and D. Sarason), Mathematical Sciences Re-search Institute Publications Vol. 33, Cambridge University Press, 1998, pp. 303–336.

[73] L. Schlesinger. Uber die Losungen gewisser linearer Differentialgleichungen als Funk-tionen der singularen Punkte. Journal fur reine und angew. Math, 129:287–294, 1905.

[74] M.P. Schutzenberger. On the definition of a family of automata. Information andControl, 4:245–270, 1961.

xvi D. Alpay and V. Vinnikov

[75] M.P. Schutzenberger. Certain elementary families of automata. Proceedings of sym-posium on mathematical theory of automata, Polytechnic Institute Brooklyn (1962),139–153.

[76] L. Schwartz. Theorie des distributions. Publications de l’Institut de Mathematiquede l’Universite de Strasbourg, No. IX-X. Nouvelle edition, entierement corrigee, re-fondue et augmentee. Hermann, Paris, 1966.

[77] J.L. Taylor. The analytic-functional calculus for several commuting operators. ActaMath., 125:1–38, 1970.

[78] J.L. Taylor. A joint spectrum for several commuting operators. J. Functional Anal-ysis, 6:172–191, 1970.

[79] V. Vinnikov. Commuting operators and function theory on a Riemann surface. InHolomorphic spaces (Berkeley, CA, 1995), pp. 445–476. Cambridge Univ. Press,Cambridge, 1998.

[80] A. L. Vol′berg, V. V. Peller, and D. V. Yakubovich. A brief excursion into the theoryof hyponormal operators. Algebra i Analiz, 2(2):1–38, 1990.

[81] D. Xia. On pure subnormal operators with finite rank self-commutators and relatedoperator tuples. Integral Equations Operator Theory, 24(1):106–125, 1996.

[82] D.V. Yakubovich. Subnormal operators of finite type. II. Structure theorems. Rev.Mat. Iberoamericana, 14(3):623–681, 1998.

[83] A. Yger. Theorie et analyse du signal. Mathematiques appliquees. Ellipses, EditionsMarketing S.A., 32 rue Bargue, Paris 15e, 1999.

Daniel Alpay and Victor VinnikovDepartment of MathematicsBen-Gurion University of the NegevBeer-Sheva, Israele-mail: dany@math.bgu.ac.ile-mail: vinnikov@math.bgu.ac.il

Operator Theory:Advances and Applications, Vol. 157, 1–60c© 2005 Birkhauser Verlag Basel/Switzerland

Functional Models for Representationsof the Cuntz Algebra

Joseph A. Ball and Victor Vinnikov

Abstract. We present a functional model, the elements of which are formalpower series in a pair of d-tuples of non-commuting variables, for a row-unitaryd-tuple of operators on a Hilbert space. The model is determined by a weight-ing matrix (called a “Haplitz” matrix) which has both non-commutative Han-kel and Toeplitz structure. Such positive-definite Haplitz matrices then serveto classify representations of the Cuntz algebra Od with specified cyclic sub-space up to unitary equivalence. As an illustration, we compute the weightingmatrix for the free atomic representations studied by Davidson and Pitts andthe related permutative representations studied by Bratteli and Jorgensen.

Mathematics Subject Classification (2000). Primary: 47A48; Secondary: 93C35.

1. Introduction

Let U be a unitary operator on a Hilbert space K and let E be a subspace of K.Define a map Φ from K to a space of formal Fourier series f(z) =

∑∞n=−∞ fnzn

by

Φ: k �→∞∑

n=−∞(PEU∗nk)zn

where PE is the orthogonal projection onto the subspace E ⊂ K. Note that Φ(k) = 0if and only if k is orthogonal to the smallest reducing subspace for U containingthe subspace E ; in particular, Φ is injective if and only if E is ∗-cyclic for U ,i.e., the smallest subspace reducing for U and containing E is the whole space K.Denote the range of Φ by L; note that we do not assume that Φ maps K into normsquare-summable series L2(T, E) = {f(z) =

∑∞n=−∞ fnzn :

∑∞n=−∞ ‖fn‖2 <∞}.

The first author is supported by NSF grant DMS-9987636; both authors are support by a grantfrom the US-Israel Binational Science Foundation.

2 J.A. Ball and V. Vinnikov

Nevertheless, we may assign a norm to elements of L so as to make Φ a coisometry:

‖Φk‖2L = ‖P(kerΦ)⊥k‖2K.

Moreover, we see that if we set Φk =∑∞

n=−∞ fnzn for a k ∈ K (so fn = PEU∗nk),then

ΦUk =∞∑

n=−∞(PEU∗n−1k)zn

=∞∑

n=−∞fn−1z

n

= z ·∞∑

n=−∞fn−1z

n−1

= z ·∞∑

n=−∞fnzn = MzΦk,

i.e., the operator U is now represented by the operator Mz of multiplication bythe variable z on the space L.

We can make this representation more explicit as follows. The standard ad-joint Φ[∗] of Φ with respect to the L2-inner product on the target domain is definedat least on polynomials:⟨

Φk,

N∑j=−N

pjzj

⟩L2

=

⟨k, Φ[∗]

⎛⎝ N∑j=−N

pjzj

⎞⎠⟩K

where we have set

Φ[∗]

⎛⎝ N∑j=−N

pjzj

⎞⎠ =N∑

j=−N

Ujpj .

Furthermore, the range Φ[∗]P of Φ[∗] acting on polynomials (where we use Pto denote the subspace of L2(T, E) consisting of trigonometric polynomials withcoefficients in E) is dense in (ker Φ)⊥, and for Φ[∗]p an element of this dense set(with p ∈ P), we have

〈ΦΦ[∗]p, ΦΦ[∗]p〉L = 〈Φ[∗]p, Φ[∗]p〉K= 〈ΦΦ[∗]p, p〉L2 .

This suggests that we set W = ΦΦ[∗] (well defined as an operator from the spaceof E-valued polynomials P to the space L(Z, E) of formal Fourier series with co-efficients in E) and define a Hilbert space LW as the closure of WP in the innerproduct

〈Wp, Wq〉LW = 〈Wp, q〉L2 .

The Toeplitz structure of W (i.e., the fact that Wi,j = PEUj−i|E depends only onthe difference i− j of the indices) implies that the operator Mz of multiplication

Functional Models 3

by z is isometric (and in fact unitary) on LW . Conversely, starting with a positivesemidefinite Toeplitz matrix [Wi,j ] with Wi,j = Wi−j , we may form a space LW

and associated unitary operator UW equal to the multiplication operator Mz actingon LW as a functional model for a unitary operator. While the space LW ingeneral consists only of formal Fourier series and there may be no bounded pointevaluations for the elements of the space, evaluation of any one of the Fouriercoefficients is a bounded operator on the space, and gives the space at least thestructure of a formal reproducing kernel Hilbert space, an L2-version of the usualreproducing kernel Hilbert spaces of analytic functions arising in many contexts;we develop this idea of formal reproducing kernel Hilbert spaces more fully in theseparate report [4].

Note that a unitary operator can be identified with a unitary representationof the circle group T or of the C∗-algebra C(T). Given any group G or C∗-algebraA, there are two natural problems: (1) classification up to unitary equivalence ofunitary representations of G or of A, and (2) classification up to unitary equivalenceof unitary representations which include the specification of a ∗-cyclic subspace.While the solution of the first problem is the loftier goal, the second problem isarguably also of interest. Indeed, there are problems in operator theory where a∗-cyclic subspace appears naturally as part of the structure; even when this is notthe case, a solution of the second problem often can be used as a stepping stoneto a solution of the first problem. In the case of G = T or A = C(T), the theory ofLW spaces solves the second problem completely: given two unitary operators U onK and U ′ on K′ with common cyclic subspace E contained in both K and K′, thenthere is a unitary operator U : K �→ K′ satisfying UU = U ′U and U |E = IE if andonly if the associated Toeplitz matrices Wi,j = PEUj−i|E and W ′

i,j = PEU ′j−i|Eare identical, and then both U and U ′ are unitarily equivalent to UW on LW withcanonical cyclic subspace W · E ⊂ LW . A little more work must be done to analyzethe dependence on the choice of cyclic subspace E and thereby solve the firstclassification problem. Indeed, if we next solve the trigonometric moment problemfor W and find a measure µ on T (with values equal to operators on E) for whichWn =

∫T

zn dµ(z), then we arrive at a representation for the original operator U asthe multiplication operator Mz on the space L2(µ). Alternatively, one can use thetheory of the Hellinger integral (see [5]) to make sense of the space of boundaryvalues of elements of LW as a certain space of vector measures (called “charts” in[5]), or one can view the space LW as the image of the reproducing kernel Hilbertspace L(ϕ) appearing prominently in work of de Branges and Rovnyak in theirapproach to the spectral theory for unitary operators (see, e.g., [6]), where

ϕ(z) =∫

T

λ + z

λ− zdµ(z) for z in the unit disk D,

under the transformation (f(z), g(z)) �→ f(z) + z−1g(z−1). In any case, the first(harder) classification problem (classification of unitary representations up to uni-tary equivalence without specification of a ∗-cyclic subspace) is solved via use ofthe equivalence relation of mutual absolute continuity on spectral measures. For

4 J.A. Ball and V. Vinnikov

this classical case, we see that the solution of the second problem serves as astepping stone to the solution of the first problem, and that the transition fromthe second to the first involves some non-trivial mathematics (e.g., solution of thetrigonometric moment problem and measure theory).

The present paper concerns representations of the Cuntz algebra Od (see,e.g., [8] for the definition and background), or what amounts to the same thing, ad-tuple of operators U = (U1, . . . ,Ud) on a Hilbert space K which is row-unitary,i.e., ⎡⎢⎣U

∗1...U∗

d

⎤⎥⎦ [U1 . . . Ud

]=

⎡⎢⎣I. . .

I

⎤⎥⎦ ,[U1 . . . Ud

]⎡⎢⎣U∗1...U∗

d

⎤⎥⎦ = I.

Equivalently, U = (U1, . . . ,Ud) is a d-tuple of isometries on K with orthogonalranges and with span of the ranges equal to the whole space K. It is known thatOd is NGCR, and hence the first classification problem for the case of Od is in-tractable in a precise sense, although particular special cases have been workedout (see [7, 9]). The main contribution of the present paper is that there is a sat-isfactory solution of the second classification problem (classification up to unitaryequivalence of unitary representations with specification of ∗-cyclic subspace) forthe case of Od via a natural multivariable analogue of the spaces LW sketchedabove for the single-variable case.

In detail, the functional calculus for a row-unitary d-tuple U = (U1, . . . ,Ud),involves the free semigroup Fd on a set of d generators {g1, . . . , gd}; elements ofthe semigroup are words w of the form w = gin . . . gi1 with i1, . . . , in ∈ {1, . . . , d}.If w = gin . . . gi1 , set Uw = Uin · · · Ui1 . The functional model for such a row-unitaryd-tuple will consist of formal power series of the form

f(z, ζ) =∑

v,w∈Fd

fv,wzvζw (1.1)

where z = (z1, . . . , zd) and ζ = (ζ1, . . . , ζd) is a pair of d non-commuting variables.The formalism is such that zizj = zjzi and ζiζj = ζjζi for i = j but ziζj = ζjzi forall i, j = 1, . . . , d. In the expression (1.1), for w = gin · · · gi1 we set zw = zin · · · zi1

and similarly for ζ. The space LW of non-commuting formal power series whichserves as the functional model for the row-unitary U = (U1, . . . ,Ud) with cyclicsubspace E will be determined by a weighting matrix

Wv,w;α,β = PEUwU∗vUα�U∗β� |Ewith row-index (v, w) and column index (α, β) in the Cartesian product Fd ×Fd. On the space LW is defined a d-tuple of generalized shift operators UW =(UW,1, . . . ,UW,d) (see formula (2.12) below) which is row-unitary and which havethe subspace W · E as a ∗-cyclic subspace. Matrices W (with rows and columnsindexed by Fd×Fd) arising in this way from a row-unitary U can be characterizedby a non-commutative analogue of the Toeplitz property which involves both anon-commutative Hankel-like and non-commutative Toeplitz-like property along

Functional Models 5

with a non-degeneracy condition; we call such matrices “Cuntz weights”. SuchCuntz weights serve as a complete unitary invariant for the second classificationproblem for the Cuntz algebraOd: given two row-unitary d-tuples U = (U1, . . . ,Ud)on K and U ′ = (U ′

1, . . . ,U ′d) on K′ with common ∗-cyclic subspace E contained in

both K and K′, then there is a unitary operator U : K �→ K′ such that UUj = U ′jU

and UU∗j = U ′∗

j U for j = 1, . . . , d and U |E = IE if and only if the associated Cuntzweights Wv,w;α,β = PEUwU∗vUα�U∗β� |E and W ′

v,w;α,β = PEU ′wU ′∗vU ′α�U ′∗β� |Eare identical, and then both U and U ′ are unitarily equivalent to the model row-unitary d-tuple UW = (UW,1, . . . ,UW,d) acting on the model space LW with canon-ical ∗-cyclic subspace W · E ⊂ LW .

The parallel with the commutative case can be made more striking by view-ing LW as a non-commutative formal reproducing kernel Hilbert space, a naturalgeneralization of classical reproducing kernel Hilbert spaces to the setting wherethe elements of the space are formal power series in a collection of non-commutingindeterminates; we treat this aspect in the separate report [4].

A second contribution of this paper is the application of this functional modelfor row-unitary d-tuples to the free atomic representations and permutative rep-resentations of Od appearing in [9] and [7] respectively. These representations areof two types: the orbit-eventually-periodic type, indexed by a triple (x, y, λ) wherex and y are words in Fd and λ is a complex number of modulus 1, and the orbit-non-periodic case, indexed by an infinite word x = gk1gk2 · · · gkn · · · . Davidson andPitts [9] have identified which pairs of parameters (x, y, λ) or x give rise to unitarilyequivalent representations of Od, which parameters correspond to irreducible rep-resentations, and how a given representation can be decomposed as a direct sum ordirect integral of irreducible representations. The contribution here is to recoverthese results (apart from the identification of irreducible representations) as anapplication of the model theory of LW spaces and the calculus of Cuntz weights.The approach shares the advantages and disadvantages of the de Branges-Rovnyakmodel theory for single operators (see [6]). Once Cuntz weights W are calculated,identifying unitary equivalences is relatively straightforward and obtaining decom-positions is automatic up to the possible presence of overlapping spaces. There issome hard work involved to verify that the overlapping space is actually trivial inspecific cases of interest. While these results are obtained in an elementary wayin [9], our results here show that a model theory calculus, a non-commutativemultivariable extension of the single-variable de Branges-Rovnyak model theory,actually does work, and in fact is straightforward modulo overlapping spaces.

The paper is organized as follows. After the present Introduction, Section2 lays out the functional models for row-isometries and row-unitary operator-tuples in particular. We show there that the appropriate analogue for a bi-infiniteToeplitz matrix is what we call a “Haplitz operator”. Just as Toeplitz operatorsW = [Wi−j ]i,j=...,−1,0,1,... have symbols W (z) =

∑∞n=−∞ Wnzn, it is shown that

associated with any Haplitz operator W is its symbol W (z, ζ), a formal power seriesin two sets of non-commuting variables (z1, . . . , zd) and ζ1, . . . , ζd). These symbols

6 J.A. Ball and V. Vinnikov

serve as the set of free parameters for the class of Haplitz operators; many questionsconcerning a Haplitz operator W can be reduced to easier questions concerningits symbol W (z, ζ). In particular, positivity of the Haplitz operator W is shown tobe equivalent to a factorization property for its symbol W (z, ζ) and for the Cuntzdefect D

W(z, ζ) of its symbol (see Theorem 2.8). Cuntz weights are characterized

as those positive semidefinite Haplitz operators with zero Cuntz defect.Section 3 introduces the analogue of L∞ and H∞, namely, the space of in-

tertwining maps LW,W∗T between two row-unitary model spaces LW and LW∗ , and

the subclass of such maps (“analytic intertwining operators”) which preserve thesubspaces analogous to Hardy subspaces. The contractive, analytic intertwiningoperators then form an interesting non-commutative analogue of the “Schur class”which has been receiving much attention of late from a number of points of view(see, e.g., [2]). These results can be used to determine when two functional modelsare unitarily equivalent, or when a given functional model decomposes as a directsum or direct integral of internal pieces (modulo overlapping spaces). Section 4gives the application of the model theory and calculus of Cuntz weights to freeatomic and permutative representations of Od discussed by Davidson and Pitts [9]and Bratteli and Jorgensen [7] mentioned above.

In a separate report [3] we use the machinery developed in this paper (es-pecially the material in Section 3) to study non-commutative analogues of Lax-Phillips scattering and unitary colligations, how they relate to each other, andhow they relate to the model theory for row-contractions developed in the workof Popescu ([12, 13, 14, 15]).

2. Models for row-isometries and row-unitaries

Let F be the free semigroup on d generators g1, . . . , gd with identity. A genericelement of Fd (apart from the unit element) has the form of a word w = gin · · · gi1 ,i.e., a string of symbols αn · · ·α1 of finite length n with each symbol αk belongingto the alphabet {g1, . . . , gd}. We shall write |w| for the length n of the wordw = αn · · ·α1. If w = αn · · ·α1 and v = βm · · ·β1 are words, then the product vwof v and w is the new word formed by the juxtaposition of v and w:

vw = βm · · ·β1αn · · ·α1.

We define the transpose w� of the word w = gin · · · gi1 by w� = gi1 · · · gin . Wedenote the unit element of Fd by ∅ (corresponding to the empty word). In partic-ular, if gk is a word of unit length, we write gkw for gkαn · · ·α1 if w = αn · · ·α1.Although Fd is a semigroup, we will on occasion work with expressions involvinginverses of words in Fd; the meaning is as follows: if w and v are words in Fd, theexpression wv−1 means w′ if there is a w′ ∈ Fd for which w = w′v; otherwise wesay that wv−1 is undefined. An analogous interpretation applies for expressionsof the form w−1v. This convention requires some care as associativity can fail: ingeneral it is not the case that (wv−1) · w′ = w · (v−1w′).

Functional Models 7

For E an auxiliary Hilbert space, we denote by �(Fd, E) the set of all E-valuedfunctions v �→ f(v) on Fd. We will write �2(Fd, E) for the Hilbert space consistingof all elements f in �(Fd, E) for which

‖f‖2�2(Fd,E) :=∑

v∈Fd

‖f(v)‖2E <∞.

Note that the space �2(Fd, E) amounts to a coordinate-dependent view of theFock space studied in [1, 9, 10, 12, 13]. It will be convenient to introduce thenon-commutative Z-transform f �→ f(z) on �(Fd, E) given by

f(z) =∑

v∈Fd

f(w)zw

where z = (z1, . . . , zd) is to be thought of as a d-tuple of non-commuting variables,and we write

zw = zin · · · zi1 if w = gin · · · gi1 .

We denote the set of all such formal power series f(z) also as L(Fd, E) (or L2(Fd, E)for the Hilbert space case). The right creation operators SR

1 , . . . , SRd on �2(Fd, E)

are given bySR

j : f �→ f ′ where f ′(w) = f(wg−1j )

with adjoint given by

SR∗j : f �→ f ′ where f ′(w) = f(wgj).

(Here f(wg−1j ) is interpreted to be equal to 0 if wg−1

j is undefined.) In the non-commutative frequency domain, these right creation operators (still denoted bySR

1 , . . . , SRd for convenience) become right multiplication operators:

SRj : f(z) �→ f(z) · zj, SR∗

j : f(z) �→ f(z) · z−1j .

In the latter expression zw · z−1j is taken to be 0 in case the word w is not of the

form w′gj for some w′ ∈ Fd. The calculus for these formal multiplication operatorsis often easier to handle; hence in the sequel we will work primarily in the non-commutative frequency-domain setting L(Fd, E) rather than in the time-domainsetting �(Fd, E).

Let K be a Hilbert space and U = (U1, . . . ,Ud) a d-tuple of operators on K.We say that U is a row-isometry if the block-operator row-matrix[

U1 · · · Ud

]: ⊕d

k=1 K �→ Kis an isometry. Equivalently, each of U1, . . . ,Ud is an isometry on K and the imagespaces imU1, . . . , imUd are pairwise orthogonal. There are two extreme cases ofrow-isometries U : (1) the case where U is row-unitary, i.e.,

[U1 . . . Ud

]is uni-

tary, or equivalently, imU1, . . . , imUd span the whole space K, and (2) the casewhere U is a row-shift, i.e.,⋂

n≥0

span{imUv : |v| = n} = {0};

8 J.A. Ball and V. Vinnikov

here we use the non-commutative multivariable operator notation

Uv = Uin . . .Ui1 if v = gin · · · gi1 .

A general row-isometry is simply the direct sum of these extreme cases by theWold decomposition for row-isometries due to Popescu (see [14]). It is well knownthat the operators SR

1 , . . . , SRd provide a model for any row-shift, as summarized

in the following.

Proposition 2.1. The d-tuple of operators (SR1 , . . . , SR

d ) on the space L2(Fd, E) isa row-shift. Moreover, if U = (U1, . . . ,Ud) is any row-shift on a space K, then Uis unitarily equivalent to (SR

1 , . . . , SRd ) on L2(Fd, E), with E = K

[⊕d

k=1UkK].

To obtain a similar concrete model for row-unitaries, we proceed as follows.Denote by �(Fd ×Fd, E) the space of all E-valued functions on Fd ×Fd:

f : (v, w) �→ f(v, w).

We denote by �2(Fd×Fd, E) the space of all elements f ∈ �(Fd×Fd, E) for which

‖f‖2�2(Fd×Fd,E) :=∑

v,w∈Fd

‖f(v, w)‖2 <∞.

The Z-transform f �→ f for elements of this type is given by

f(z, ζ) =∑v,w

f(v, w)zvζw.

Here z = (z1, . . . , zd) is a d-tuple of non-commuting variables as before, and ζ =(ζ1, . . . , ζd) is another d-tuple of non-commuting variables, but we specify thateach ζi commutes with each zj for i, j = 1, . . . , d. For the case d = 1, note that�2(F1, E) is the standard �2-space over the non-negative integers �2(Z+, E), while

�2(F1 ×F1, E) = �2(Z+ × Z+, E)appears to be a more complicated version of �2(Z, E). Nevertheless, we shall seethat the weighted modifications of �2(Fd×Fd, E) which we shall introduce below docollapse to �2(Z, E) for the case d = 1. Similarly, one should think of L2(Fd, E) as anon-commutative version of the Hardy space H2(D, E) over the unit disk D, and ofthe modifications of L2(Fd×Fd, E) to be introduced below as a non-commutativeanalogue of the Lebesgue space L2(T, E) of measurable norm-square-integrableE-valued functions on the unit circle T.

In the following we shall focus on the frequency domain setting L2(Fd ×Fd, E) rather than the time-domain setting �2(Fd,×Fd, E), where it is convenientto use non-commutative multiplication of formal power series; for this reason weshall write simply f(z, ζ) for elements of the space rather than f(z, ζ). Unlike theunilateral setting L2(Fd, E) discussed above, there are two types of shift operatorson L2(Fd ×Fd, E) of interest, namely:

SRj : f(z, ζ) �→ f(z, ζ) · zj , (2.1)

URj : f(z, ζ) �→ f(0, ζ) · ζ−1

j + f(z, ζ) · zj (2.2)

Functional Models 9

where f(0, ζ) is the formal power series in ζ = (ζ1, . . . , ζd) obtained by formallysetting z = 0 in the formal power series for f(z, ζ):

f(0, ζ) =∑

w∈Fd

f∅,wζw if f(z, ζ) =∑

v,w∈Fd

fv,wzvζw.

One can think of SRj as a non-commutative version of a unilateral shift (even

in this bilateral setting), while URj is some kind of bilateral shift. We denote by

SR[∗]j and U

R[∗]j the adjoints of SR

j and URj in the L2(Fd × Fd, E)-inner product

(to avoid confusion with the adjoint with respect to a weighted inner product toappear below). An easy computation shows that

SR[∗]j : f(z, ζ) �→ f(z, ζ) · z−1

j , (2.3)

UR[∗]j : f(z, ζ) �→ f(0, ζ) · ζj + f(z, ζ) · z−1

j . (2.4)

Note that

UR[∗]i SR

j : f(z, ζ) �→ UR[∗]i (f(z, ζ) · zj) = δi,jf(z, ζ)

and hence we have the useful identity

UR[∗]i SR

j = δi,jI. (2.5)

On the other hand

SRj U

R[∗]j : f(z, ζ) �→SR

j (f(0, ζ)ζj + f(z, ζ)z−1j )

= f(0, ζ)ζjzj + [f(z, ζ)z−1j ]zj

and hence⎛⎝I −d∑

j=1

SRj U

R[∗]j

⎞⎠ : f(z, ζ) �→f(z, ζ)−d∑

j=1

f(0, ζ)ζjzj −d∑

j=1

[f(z, ζ)z−1j ]zj

= f(0, ζ)−d∑

j=1

f(0, ζ)ζjzj

and hence ⎛⎝I −d∑

j=1

SRj U

R[∗]j

⎞⎠ : f(z, ζ) �→ f(0, ζ) ·

⎛⎝1−d∑

j=1

zjζj

⎞⎠ . (2.6)

Now suppose that U = (U1, . . . ,Ud) is a row-unitary d-tuple of operators on aHilbert spaceK, E is a subspace ofK, and we define a map Φ: K �→ L(Fd×Fd, E) by

Φk =∑

v,w∈Fd

(PEUwU∗vk)zvζw. (2.7)

10 J.A. Ball and V. Vinnikov

Then

Φ ◦ Ujk =∑

v,w∈Fd

(PEUwU∗vUjk)zvζw

=∑

w∈Fd

(PEUwgj k)ζw +∑

v,w : v =∅(PEUwU∗vg−1

j k)zvζw

= (URj ◦ Φk)(z, ζ) (2.8)

while

Φ ◦ U∗j k =

∑v,w∈Fd

(PEUvU∗wU∗j k)zvζw

=∑

v,w∈Fd

(PEUvU∗wgj k)zvζw

= (SR[∗]j ◦ Φk)(z, ζ). (2.9)

If we let W = ΦΦ[∗] (where Φ[∗] is the adjoint of Φ with respect to the Hilbertspace inner product on K and the formal L2-inner product on L(Fd×Fd, E)), then

Φ[∗] : zαζβe �→ Uα�U∗β�e

and W := ΦΦ[∗] = [Wv,w;α,β]v,w,α,β∈Fdwhere

Wv,w;α,β = PEUwU∗vUα�Uβ� |E . (2.10)

If imΦ is given the lifted norm ‖ · ‖�,‖Φk‖� = ‖P(kerΦ)⊥k‖K ,

then one easily checks that W · P(Fd ×Fd, E) ⊂ imΦ and

‖Wp‖2� = ‖Φ[∗]p‖2K= 〈Φ[∗]p, Φ[∗]p〉K= 〈Wp, p〉L2 .

Thus, if we define a space LW as the closure of W · P(Fd ×Fd, E) in the norm

‖Wp‖2LW= 〈Wp, p〉L2 , (2.11)

then LW = imΦ isometrically. From the explicit form (2.10) of Wv,w;α,β it is easyto verify the intertwining relations

URj W = WSR

j , SR[∗]j W = WU

R[∗]j on P(Fd ×Fd, E).

If we define UW = (UW,1, . . . ,UW,d) on LW by

UW,j : Wp �→ URj Wp = WSjp, (2.12)

then, from the intertwining relations

ΦUj = URj Φ, ΦU∗

j = SR[∗]j Φ for j = 1, . . . , d

Functional Models 11

and the fact that Φ is coisometric, we deduce that UW is row-unitary on LW withadjoint U∗

J = (U∗W,1, . . . ,U∗

W,d) on LW given by

U∗W,j : Wp �→ S

R[∗]j Wp = WU

R[∗]j p. (2.13)

If Φ is injective (or, equivalently, if E is ∗-cyclic for the row-unitary d-tupleU = (U1, . . . ,Ud)), then UW = (UW,1, . . . ,UW,d) on LW is a functional modelrow-unitary d-tuple for the abstractly given row-unitary d-tuple U = (U1, . . .Ud).

Our next goal is to understand more intrinsically which weights

W = [Wv,w;α,β]

can be realized in this way as (2.10) and thereby lead to functional models forrow-unitary d-tuples U . From identity (2.5) we see that (SR

1 , . . . , SRd ) becomes a

row-isometry if we can change the inner product on L2(Fd × Fd, E) so that theadjoint SR∗

j of Sj in the new inner product is UR[∗]j . Moreover, if we in addition

arrange for the new inner product to have enough degeneracy to guarantee thatall elements of the form f(0, ζ)(1 −∑d

j=1 zjζj) have zero norm, then the d-tuple(SR

1 , . . . , SRd ) in this new inner product becomes row-unitary. These observations

suggest what additional properties we seek for a weight W so that it may be ofthe form (2.10) for a row-unitary U .

Let W be a function from four copies of Fd into bounded linear operators onE with value at (v, w, α, β) denoted by Wv,w;α,β. We think of W as a matrix withrows and columns indexed by Fd ×Fd; thus Wv,w;α,β is the matrix entry for row(v, w) and column (α, β). Denote by P(Fd,×Fd, E) the space of all polynomials inthe non-commuting variables z1, . . . , zd, ζ1, . . . , ζd:

P(Fd,×Fd, E) ={p(z, ζ) =∑

v,w∈Fd

pv,wzvζw : pv,w ∈ E and

pv,w = 0 for all but finitely many v, w}.

Then W can be used to define an operator from P(Fd×Fd, E) into L(Fd×Fd, E)by extending the formula

W : e zαζβ �→∑

v,w∈Fd

Wv,w;α,βe zvζw .

for monomials to all of P(Fd,×Fd, E) by linearity. Note that computation of theL2-inner product 〈Wp, q〉L2 involves only finite sums if p and q are polynomials,and therefore is well defined. We say that W is positive semidefinite if

〈Wp, p〉L2 ≥ 0 for all p ∈ P(Fd ×Fd, E).

Under the assumption that W is positive semidefinite, define an inner product onW · P(Fd ×Fd, E) by

〈Wp, Wq〉LW = 〈Wp, q〉L2 . (2.14)

12 J.A. Ball and V. Vinnikov

Modding out by elements of zero norm if necessary, W · P(Fd × Fd, E) is a pre-Hilbert space in this inner product. We define a space

LW = the completion of W · P(Fd × Fd, E) in the inner product (2.14). (2.15)

Note that the (v, w)-coefficient of Wp ∈ WP(Fd ×Fd, E) is given by

〈[Wp]v,w, e〉E = 〈Wp, W (zvζwe)〉LW (2.16)

and hence the map Φv,w : f �→ fv,w extends continuously to the completion LW

of W · P(Fd × Fd, E) and LW can be identified as a space of formal power seriesin the non-commuting variables z1, . . . , zd and ζ1, . . . , ζd, i.e., as a subspace ofL(Fd×Fd, E). (This is the main advantage of defining the space as the completionof W · P(Fd×Fd, E) rather than simply as the completion of P(Fd×Fd, E) in theinner product given by the right-hand side of (2.14).) Note that then, for f ∈ LW

and α, β ∈ FD we have

〈Φα,βf, e〉E = 〈fα,β , e〉E = 〈f, W [zαζβe]〉LW

from which we see that Φ∗α,β : E �→ LW is given by

Φ∗α,β : e �→W [zαζβe].

By using this fact we see that

〈Φv,wΦ∗α,βe, e′〉E = 〈Φ∗

α,βe, Φ∗v,we′〉LW

= 〈W [zαζβe], W [zvζwe′]〉LW

= 〈W [zαζβe], zvζwe′〉LW

= 〈W [zαζβe], zvζwe′〉L2

= 〈Wv,w;α,βe, e′〉Eand we recover the operator matrix entries Wv,w;α,β of the operator W from thefamily of operators Φα,β (α, β ∈ Fd) via the factorization

Wv,w;α,β = Φv,wΦ∗α,β .

Conversely, one can start with any such factorization of W (through a generalHilbert space K rather than K = LW as in the construction above). The followingtheorem summarizes the situation.

Theorem 2.2. Assume that W = [Wv,w;α,β ]v,w,α,β∈Fdis a positive semidefinite

(Fd×Fd)×(Fd×Fd) matrix of operators on the Hilbert space E with a factorizationof the form

Wv,w;α,β = Φv,wΦ∗α,β

for operators Φv,w : K �→ E for some intermediate Hilbert space K for all v, w, α,β ∈ Fd. Define an operator Φ: K �→ L(Fd ×Fd, E) by

Φ: k �→∑

v,w∈Fd

(Φv,wk)zvζw .

Functional Models 13

Let LW be the Hilbert space defined as the completion of WP(Fd × Fd, E) in thelifted inner product

〈Wp, Wp〉LW = 〈Wp, p〉L2 .

Then Φ is a coisometry from K onto LW with adjoint given densely by

Φ∗ : Wp �→∑

v,w∈Fd

Φ∗α,βpα,β

for p(z, ζ) =∑

α,β pα,βzαζβ a polynomial in P(Fd × Fd, E). In particular, LW isgiven more explicitly as LW = imΦ.

Proof. Note than W (as a densely defined operator on L2(Fd×Fd, E) with domaincontaining at least P(Fd ×Fd, E)) factors as W = ΦΦ[∗], where Φ[∗] is the formalL2-adjoint of Φ defined at least on polynomials by

Φ[∗] : p(z, w) �→∑α,β

Φ∗α,βpα,β for p(z, w) =

∑α,β

pα,βzαζβ .

In particular, Φ[∗]P(Fd × Fd, E) is contained in the domain of Φ when Φ is con-sidered as an operator from K into LW with domΦ = {k ∈ K : Φk ∈ LW }. SinceΦ is defined in terms of matrix entries Φv,w and evaluation of Fourier coefficientsp �→ pv,w is continuous in LW , it follows that Φ as an operator from K into LW

with domain as above is closed. For an element k ∈ K of the form Φ[∗]p for apolynomial p ∈ P(Fd ×Fd, E), we have

〈Wp, Wp〉LW = 〈Wp, p〉L2

= 〈ΦΦ[∗]k, k〉L2

= 〈Φ[∗]k, Φ[∗]k〉K.

Hence Φ maps Φ[∗]P(Fd × Fd, E) isometrically onto the dense submanifold W ·P(Fd ×Fd, E) of LW . From the string of identities

〈k, Φ[∗]p〉K = 〈Φk, p〉L2

= 〈Φk, Wp〉LW

and the density of W · P(Fd × Fd, E) in LW , we see that kerΦ = (Φ[∗]P(Fd ×Fd, E))⊥. Hence Φ is isometric from a dense subset of the orthogonal complement ofits kernel (Φ[∗]P(Fd×Fd, E)) onto a dense subset of LW (namely, WP(Fd×Fd, E)).Since Φ is closed, it follows that necessarily Φ: K �→ LW is a coisometry. Finally,notice that

〈Φk, Wp〉LW = 〈Φk, p〉L2

= 〈k, Φ[∗]p〉Kfrom which we see that

Φ∗ : Wp �→ Φ[∗]p for p ∈ P(Fd ×Fd, E)and the formula for Φ∗ follows. This completes the proof of Theorem 2.2. �

14 J.A. Ball and V. Vinnikov

We seek to identify additional properties to be satisfied by W so that theoperators UW,j defined initially only on W · P(Fd ×Fd, E) by

UW,j : Wp �→ WSRj p

and then extended to all of LW by continuity become a row-isometry, or even arow-unitary operator-tuple. From (2.5), the row-isometry property follows if it canbe shown that U∗

W,j : Wp �→WUR[∗]j p, or equivalently,

WSRj = UR

j W on P(Fd ×Fd, E). (2.17)

Similarly, from (2.6) we see that the row-unitary property will follow if we showin addition that

W

[p(0, ζ)

(1−

d∑k=1

zkζk

)]= 0 for all p ∈ P(Fd ×Fd, E). (2.18)

The next theorem characterizes those operator matrices W for which (2.17) and(2.18) hold.

Theorem 2.3. Let W be a (Fd ×Fd)× (Fd ×Fd) matrix with matrix entries equalto operators on the Hilbert space E. Then:

1. W satisfies (2.17) if and only if

W∅,w;αgj ,β = W∅,wgj ;α,β, (2.19)

Wv,w;αgj ,β = Wvg−1j ,w;α,β for v = ∅ (2.20)

for all v, w, α, β ∈ Fd, j = 1, . . . , d, where we interpret Wvg−1j ,w;α,β to be 0

in case vg−1j is not defined.

2. Assume that W is selfadjoint. Then W satisfies (2.17) and (2.18) if and onlyif W satisfies (2.19), (2.20) and in addition

W∅,w;∅,β =d∑

j=1

W∅,wgj ;∅,βgj(2.21)

for all w, β ∈ Fd.

Proof. By linearity, it suffices to analyze the conditions (2.17) and (2.18) on mono-mials f(z, ζ) = zαζβe for some α, β ∈ Fd and e ∈ E . We compute

WSRj (zαζβe) = W (zαgj ζβe)

=∑

v,w∈Fd

Wv,w;αgj ,βzvζw (2.22)

Functional Models 15

while, on the other hand,

URj W (zαζβe) = UR

j

⎛⎝ ∑v,w∈Fd

Wv,w;α,βezvζw

⎞⎠=∑

w∈Fd

W∅,w;α,βζwg−1j +

∑v,w∈Fd

Wv,w;α,βzvgj ζw . (2.23)

Equality of (2.22) with (2.23) for all α, β and e in turn is equivalent to (2.19) and(2.20).

Assume now that W satisfies (2.19) and is selfadjoint. Then we have

Wv,w;∅,βgj= [W∅,βgj ;v,w]∗

= [W∅,β;vgj ,w]∗

= Wvgj ,w;∅,β

and henceWvgj ,w;∅,β = Wv,w;∅,βgj

. (2.24)

To verify the condition for (2.18), if f(z, ζ) = zαζβe, then f(0, ζ) = 0 ifα = ∅ and W

(f(0, ζ)

(1−∑d

j=1 zjζj

))= 0 trivially. Thus it suffices to consider

the case α = ∅ and f(z, ζ) = ζβe. Then f(0, ζ) = ζβe and

f(0, ζ)

⎛⎝1−d∑

j=1

zjζj

⎞⎠ = ζβe−d∑

j=1

zgj ζβgj e.

Then

W

⎡⎣f(0, ζ)

⎛⎝1−d∑

j=1

zjζj

⎞⎠⎤⎦ = W

⎡⎣ζβe−d∑

j=1

zjζβgj e

⎤⎦=∑

v,w∈Fd

⎛⎝Wv,w;∅,β −d∑

j=1

Wv,w;gj ,βgj

⎞⎠ e zvζw

This last quantity set equal to zero for all β ∈ Fd and e ∈ E is equivalent to

Wv,w;∅,β =d∑

j=1

Wv,w;gj ,βgj . (2.25)

If v = ∅, we may use (2.19) to see that

W∅,w;gj ,βgj= W∅,wgj ;∅,βgj

and (2.25) collapses to (2.21). If v = ∅, write v = v′gk for some k. From (2.20) wehave

Wv,w;gj ,βgj = δj,kWv′,w;∅,βgk