operator theory systems theory and scattering theory multidimensional generalizations

8/12/2019 Operator Theory Systems Theory and Scattering Theory Multidimensional Generalizations

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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. Bttcher (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)


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Birkhuser VerlagBasel .Boston .Berlin

Operator Theory, Systems Theory

and Scattering Theory:

Multidimensional Generalizations

Daniel AlpayVictor VinnikovEditors


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Library of Congress, Washington D.C., USA

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ISBN 3-7643-7212-5 Birkhuser Verlag, Basel Boston Berlin

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specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms

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2005 Birkhuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland

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9 8 7 6 5 4 3 2 1 www.birkhauser.ch

Editors:

Daniel Alpay

Victor Vinnikov

Department of Mathematics

Ben-Gurion University of the Negev

P.O. Box 653

Beer Sheva 84105

Israel

e-mail: [email protected]

[email protected]

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


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

S. Eidelman and Y. KrasnovOperator Methods for Solutions of PDEsBased on Their Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

D.S. Kalyuzhny-VerbovetzkiOn the Bessmertny 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. LunaElizarraras and M. ShapiroPreservation of the Norms of Linear Operators Actingon some Quaternionic Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

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

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

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

F.H. SzafraniecFavards Theorem Modulo an Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301


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Operator Theory:Advances and Applications, Vol. 157, viixvic 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 quils suscitent: autant de difficultes dont

sentiche lamateur dInsoluble.

(Cioran, La tentation dexister, [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 worksinclude 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 LaxPhillips 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 BochnerChandra-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 Schwartzkernel 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, their

various incarnations and related problems, yet it is basically true that there is onlyone underlying theory.


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One natural extension of the single variable theory is the time varying case, whereone (roughly speaking) replaces the complex numbers by diagonal operators and

the 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 FornasiniMarchesini model in the multi-dimensional system theory literature; see [71], [46].

Function theory on the free semigroup and on the unit ball of CN. Fromthe 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 results

on the theory of multipliers in the so-called Arveson space of the ball (see[12]), which are closely related with the theory of complete NevanlinnaPickkernels; 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 of

the 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 Krens spectral shift function; see [62], [27], [28]. Another


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closely related topic is the study of multi-dimensional moment problems; ofthe vast literature we mention (in addition to [68]) the works of Curto and

Fialkow [33], [34] and of Putinar and Vasilescu [69]. Hyperanalytic functions and applications. Left (resp. right) hyperanalyticfunctions are quaternionic-valued functions in the kernel of the left (resp.right) CauchyFueter operator (these are extensions to R4 of the operatorx +i

y ). The theory is non-commutative and a supplementary difficulty

is 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 CauchyKovalevskaya theorem allows to associate (at least locally) toany such function a hyperanalytic function. Identifying the quaternions with

C2 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 zn11 z

n22 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 tostudy 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 Bolibruchs coun-terexample to Hilberts 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 Bessmertny, 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 HilbertSchmidt operator functions with symmetries on the bidisk, deals withinterpolation in the bidisk in the setting ofH2 rather than ofH.

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


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paper of Banks, Constantinescu and Johnson, Relations on non-commutative vari-ables and associated orthogonal polynomials, where the authors survey various

settings 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 Kalyuzhny-Verbovetzki, On the Bessmertny class of homogeneous positiveholomorphic functions on a product of matrix halfplanes, a recent investigationof the author on the Bessmertny 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 generalnon-commutative domain, a product of matrix halfplanes. The study of the Bess-

mertny class (as well as its generalization) is motivated by the electrical networkstheory: as shown by M.F. Bessmertny [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 of

the 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


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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 representations

of Clifford-analytic functions in terms of power series in operator indeterminates.LunaElizarraras and Shapiro inPreservation 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 holomorphic

families 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 ofthem 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 the

joint 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 Hilbertmodules; see [42], [40], [41].

The work of Agler, Young and their collaborators on operator theory andrealization theory related to function theory on the symmetrized bidisk, withapplications to the two-by-two spectral NevanlinnaPick problem; see [5], [4],[6].

Spectral analysis and the notion of the characteristic function for commutingoperators, 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, withapplications 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 theory

and scattering theory: multi-dimensional generalizations, 2003, which was held atthe Department of Mathematics of Ben-Gurion University of the Negev during theperiod June 30July 3, 2003. It is a pleasure to thank all the participants for an


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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 the

workshop possible.

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[63] G. Popescu. Characteristic functions for infinite sequences of noncommuting opera-tors.J. Operator Theory, 22(1):5171, 1989.

[64] G. Popescu. Isometric dilations for infinite sequences of noncommuting operators.

Trans. Amer. Math. Soc., 316(2):523536, 1989.

[65] G. Popescu. Models for infinite sequences of noncommuting operators. Acta Sci.Math. (Szeged), 53(3-4):355368, 1989.

[66] G. Popescu. Multi-analytic operators on Fock spaces. Math. Ann., 303(1):3146,1995.

[67] S. Pott. Standard models under polynomial positivity conditions. J. Operator The-ory, 41:365389, 1999.

[68] M. Putinar. Extremal solutions of the two-dimensional L-problem of moments. J.Funct. Anal., 136(2):331364, 1996.

[69] M. Putinar and F.-H. Vasilescu. Solving moment problems by dimensional extension.Ann. of Math. (2) 149 (1999), no. 3, 10871107.

[70] P. Quiggin. For which reproducing kernel Hilbert spaces is Picks theorem true?Integral Equations Operator Theory, 16:244266, 1993.

[71] R. Roesser. A discrete state-space model for linear image processing. IEEE Trans.Automatic Control, AC20:110, 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. 303336.

[73] L. Schlesinger. Uber die Losungen gewisser linearer Differentialgleichungen als Funk-

tionen der singularen Punkte.Journal fur reine und angew. Math, 129:287294, 1905.[74] M.P. Schutzenberger. On the definition of a family of automata. Information and

Control, 4:245270, 1961.


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

139153.[76] L. Schwartz. Theorie des distributions. Publications de lInstitut de Mathematique

de lUniversite 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., 125:138, 1970.

[78] J.L. Taylor. A joint spectrum for several commuting operators. J. Functional Anal-ysis, 6:172191, 1970.

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Daniel Alpay and Victor VinnikovDepartment of MathematicsBen-Gurion University of the NegevBeer-Sheva, Israele-mail: [email protected]: [email protected]


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Operator Theory:Advances and Applications, Vol. 157, 160c 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 ofd-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 Ube a unitary operator on a Hilbert spaceK and letEbe a subspace ofK.Define a map fromK to a space of formal Fourier series f(z) =

n= fnz

n

by

: k

n=(PEUnk)zn

where PEis the orthogonal projection onto the subspace E K. Note that (k) = 0if and only ifk is orthogonal to the smallest reducing subspace forU containingthe subspaceE; in particular, is injective if and only ifE is-cyclic forU,i.e., the smallest subspace reducing forU and containingE is the whole spaceK.Denote the range of by L; note that we do not assume that maps K into normsquare-summable series L2(T, E) ={f(z) =

n= fnz

n :

n= fn2


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2 J.A. Ball and V. Vinnikov

Nevertheless, we may assign a norm to elements ofL so as to make a coisometry:

k

2

L=

P(ker)k

2

K.

Moreover, we see that if we set k= n= fnzn for ak K(sofn=PEUnk),then

Uk=

n=(PEUn1k)zn

=

n=fn1zn

=z

n= fn1zn1=z

n=

fnzn =Mzk,

i.e., the operatorU is now represented by the operator Mz of multiplication bythe variablez on the spaceL.

We can make this representation more explicit as follows. The standard ad-joint [] of with respect to theL2-inner product on the target domain is definedat least on polynomials:

k,

Nj=N

pjzjL2

=

k, [] Nj=N

pjzj

Kwhere we have set

[]

Nj=N

pjzj

= Nj=N

Ujpj .

Furthermore, the range []P of [] acting on polynomials (where we usePto denote the subspace ofL2(T, E) consisting of trigonometric polynomials withcoefficients inE) is dense in (ker ), and for

[]

p an element of this dense set(with p P), we have

[]p, []pL= []p, []pK= []p, pL2.

This suggests that we set W = [] (well defined as an operator from the spaceofE-valued polynomialsP to the space L(Z, E) of formal Fourier series with co-efficients inE) and define a Hilbert spaceLWas the closure ofWP in the innerproduct

W p , W q LW = Wp,qL2

.The Toeplitz structure ofW(i.e., the fact thatWi,j =PEUji|Edepends only onthe difference i j of the indices) implies that the operator Mz of multiplication


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Functional Models 3

byz is isometric (and in fact unitary) on LW. Conversely, starting with a positivesemidefinite Toeplitz matrix [Wi,j ] with Wi,j =Wij , we may form a spaceLWand associated unitary operatorUWequal to the multiplication operator MzactingonLW as a functional model for a unitary operator. While the spaceLW 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 representation

of the circle groupT or of theC-algebraC(T). Given any groupG or C-algebraA, there are two natural problems: (1) classification up to unitary equivalence ofunitary representations ofG or ofA, 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 ofG = Tor A= C(T), the theory of

LWspaces solves the second problem completely:given two unitary operators

Uon

K andU onK with common cyclic subspaceEcontained in bothK andK, thenthere is a unitary operatorU:K K satisfyingUU=UU andU|E=IE if andonly if the associated Toeplitz matrices Wi,j = PEUji|E and Wi,j = PEUji|Eare identical, and then bothU andU are unitarily equivalent toUW onLW withcanonical cyclic subspaceW E LW.A little more work must be done to analyzethe dependence on the choice of cyclic subspaceE and thereby solve the firstclassification problem. Indeed, if we next solve the trigonometric moment problemfor Wand find a measure onT (with values equal to operators onE) for whichWn = T

zn d(z), then we arrive at a representation for the original operatorUasthe multiplication operatorMz on the spaceL

2

(). Alternatively, one can use thetheory of the Hellinger integral (see [5]) to make sense of the space of boundaryvalues of elements ofLWas a certain space of vector measures (called charts in[5]), or one can view the spaceLWas the image of the reproducing kernel HilbertspaceL() appearing prominently in work of de Branges and Rovnyak in theirapproach to the spectral theory for unitary operators (see, e.g., [6]), where

(z) =

T

+ z

z d(z) forz in the unit disk D,

under the transformation (f(z), g(z)) f(z) +z1g(z1). 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


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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 from

the 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 algebraOd (see,e.g., [8] for the definition and background), or what amounts to the same thing, ad-tuple of operatorsU = (U1, . . . ,Ud) on a Hilbert spaceK which is row-unitary,i.e.,

U1...

Ud

U1 . . . Ud =

I

. . .

I

,

U1 . . . Ud

U1...

Ud

=I .

Equivalently,U = (U1, . . . ,Ud) is a d-tuple of isometries onK with orthogonalranges and with span of the ranges equal to the whole spaceK. It is known thatOd is NGCR, and hence the first classification problem for the case ofOd 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 ofOd via a natural multivariable analogue of the spacesLW sketchedabove for the single-variable case.

In detail, the functional calculus for a row-unitaryd-tupleU= (U1, . . . ,Ud),involves the free semigroupFd on a set ofd generators{g1, . . . , gd}; elements ofthe semigroup are wordsw of the form w= gin. . . gi1 with i1, . . . , in {1, . . . , d}.Ifw = gin. . . gi1 , setUw =Uin U i1 . The functional model for such a row-unitaryd-tuple will consist of formal power series of the form

f(z, ) =

v,wFdfv,wz

vw (1.1)

wherez = (z1, . . . , zd) and= (1, . . . , d) is a pair ofd non-commutingvariables.The formalism is such that zizj=zjzi andij=ji fori =j butzij =jziforalli, j= 1, . . . , d. In the expression (1.1), forw= gin gi1 we setz

w

=zin zi1and similarly for . The spaceLWof non-commuting formal power series whichserves as the functional model for the row-unitaryU = (U1, . . . ,Ud) with cyclicsubspaceEwill be determined by a weighting matrix

Wv,w;, =PEUwUvUU |Ewith row-index (v, w) and column index (, ) in the Cartesian productFdFd. On the spaceLW is defined a d-tuple of generalized shift operatorsUW =(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 FdFd) arising in this way from a row-unitaryUcan be characterizedby a non-commutative analogue of the Toeplitz property which involves both anon-commutative Hankel-like and non-commutative Toeplitz-like property along


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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 classification

problem for the Cuntz algebra Od:given two row-unitaryd-tuplesU= (U1, . . . ,Ud)onK andU = (U1, . . . ,Ud) onK with common-cyclic subspaceEcontained inbothK andK, then there is a unitary operatorU:K K such thatUUj =UjUandUUj =Uj U forj = 1, . . . , dandU|E=IEif and only if the associated Cuntzweights Wv,w;, = PEUwUvUU|E and Wv,w;, = PEUwUvU

U |Eare identical, and then bothU andU are unitarily equivalent to the model row-unitaryd-tupleUW = (UW,1, . . . ,UW,d) acting on the model spaceLW with canon-ical-cyclic subspaceW E LW.

The parallel with the commutative case can be made more striking by view-

ingLW 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 ofOd appearing in [9] and [7] respectively. These representations areof two types: theorbit-eventually-periodictype, indexed by a triple (x,y,) wherexand y are words inFd andis a complex number of modulus 1, and the orbit-non-periodiccase, indexed by an infinite wordx = gk1gk2

gkn

. Davidson and

Pitts [9] have identified which pairs of parameters (x,y,) or xgive rise to unitarilyequivalent representations ofOd, 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 ofLWspaces 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 Ware calculated,identifying unitary equivalences is relatively straightforward and obtaining decom-positions is automatic up to the possible presence of overlapping spaces. There is

some 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 operators

W = [Wij ]i,j=...,1,0,1,... have symbolsW(z) = n= Wnzn, it is shown thatassociated with any Haplitz operator Wis its symbolW(z, ), a formal power seriesin two sets of non-commuting variables (z1, . . . , zd) and1, . . . , d). These symbols


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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 concerning

its symbolW(z, ). In particular, positivity of the Haplitz operatorWis shown tobe equivalent to a factorization property for its symbolW(z, ) and for the CuntzdefectDW(z, ) of its symbol (see Theorem 2.8). Cuntz weights are characterizedas those positive semidefinite Haplitz operators with zero Cuntz defect.

Section 3 introduces the analogue ofL and H, namely, the space of in-tertwining mapsLW,WT between two row-unitary model spacesLW andLW , andthe 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 classwhich 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 ofOd 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, and

how 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

LetF be the free semigroup on d generators g1, . . . , gd with identity. A genericelement ofFd(apart from the unit element) has the form of a word w =gin gi1 ,i.e., a string of symbols n 1 of finite lengthnwith each symbol k belongingto the alphabet{g1, . . . , gd}. We shall write|w| for the length n of the wordw=n 1. Ifw=n 1 and v =m 1 are words, then the product vwofv and w is the new word formed by the juxtaposition ofv andw:

vw = m 1n 1.We define the transpose w of the word w = gin gi1 by w = gi1 gin . Wedenote the unit element ofFd by(corresponding to the empty word). In partic-ular, ifgk is a word of unit length, we write gkw for gkn 1 ifw=n 1.AlthoughFd is a semigroup, we will on occasion work with expressions involvinginverses of words inFd; the meaning is as follows: ifwandv are words inFd, theexpressionwv1 meansw if there is a w Fd for which w=wv; otherwise wesay that wv

1

is undefined. An analogous interpretation applies for expressionsof the formw1v. This convention requires some care as associativity can fail: ingeneral it is not the case that (wv1) w =w (v1w).


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Functional Models 7

For Ean auxiliary Hilbert space, we denote by(Fd, E) the set of all E-valuedfunctionsv f(v) onFd. We will write2(Fd, E) for the Hilbert space consistingof all elements f in(Fd, E) for which

f22(Fd,E):=vFd

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 the

non-commutative Z-transformff(z) on(Fd, E) given by

f(z) =

vFd

f(w)zw

wherez = (z1, . . . , zd) is to be thought of as ad-tuple of non-commuting variables,and we write

zw =zin zi1 ifw= gin gi1 .We denote the set of all such formal power seriesf(z) also asL(Fd, E) (orL2(Fd, E)for the Hilbert space case). The right creation operators SR1, . . . , S

Rd on

2(Fd, E)are given by

SRj : f f wheref(w) =f(wg1j )with adjoint given by

SRj : f

f where f(w) =f(wgj).

(Here f(wg1j ) is interpreted to be equal to 0 ifwg1j is undefined.) In the non-

commutative frequency domain, these right creation operators (still denoted bySR1, . . . , S

Rd for convenience) become right multiplication operators:

SRj :f(z) f(z) zj, SRj :f(z) f(z) z1j .

In the latter expression zw z1j is taken to be 0 in case the wordw is not of theformwgj for somew 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-domain

setting(Fd, E).LetK be a Hilbert space andU = (U1, . . . ,Ud) a d-tuple of operators onK.

We say thatU is a row-isometry if the block-operator row-matrixU1 U d : dk=1 K Kis an isometry. Equivalently, each ofU1, . . . ,Ud is an isometry onKand the imagespaces imU1, . . . , imUd are pairwise orthogonal. There are two extreme cases ofrow-isometriesU: (1) the case whereU is row-unitary, i.e., U1 . . . Ud is uni-tary, or equivalently, imU1, . . . , imUd span the whole spaceK, and (2) the casewhere

U is a row-shift, i.e.,

n0span{imUv :|v|= n} = {0};


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here we use the non-commutative multivariable operator notation

Uv =

Uin. . .

Ui1 ifv=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 SR1, . . . , S

Rd provide a model for any row-shift, as summarized

in the following.

Proposition 2.1. The d-tuple of operators (SR1, . . . , S Rd) on the space L

2(Fd, E) isa row-shift. Moreover, ifU = (U1, . . . ,Ud) is any row-shift on a spaceK, thenUis unitarily equivalent to (SR1, . . . , S

Rd) onL

2(Fd, E), withE= K dk=1UkK .

To obtain a similar concrete model for row-unitaries, we proceed as follows.

Denote by(Fd Fd, E) the space of allE-valued functions onFd Fd:f: (v, w) f(v, w).

We denote by 2(FdFd, E) the space of all elements f (FdFd, E) for whichf22(FdFd,E):=

v,wFd

f(v, w)2 < .

TheZ-transformfffor elements of this type is given by

f(z, ) =

v,wf(v, w)zvw.

Here z = (z1, . . . , zd) is a d-tuple of non-commuting variables as before, and =(1, . . . , d) is another d-tuple of non-commuting variables, but we specify thateach i commutes with each zj for i, j = 1, . . . , d. For the case d = 1, note that2(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 of2(Z, E). Nevertheless, we shall seethat the weighted modifications of2(FdFd, E) which we shall introduce below docollapse to2(Z, E) for the cased = 1. Similarly, one should think ofL2(Fd, E) as anon-commutative version of the Hardy spaceH2(D,

E) over the unit disk D, and of

the modifications ofL2(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 circleT.

In the following we shall focus on the frequency domain setting L2(FdFd, 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 we

shall write simply f(z, ) for elements of the space rather thanf(z, ). Unlike theunilateral settingL2(Fd, E) discussed above, there are two types of shift operatorson L2(Fd Fd, E) of interest, namely:

SRj : f(z, ) f(z, ) zj , (2.1)URj :f(z, ) f(0, ) 1j + f(z, ) zj (2.2)


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Functional Models 9

where f(0, ) is the formal power series in = (1, . . . , d) obtained by formallysettingz= 0 in the formal power series for f(z, ):

f(0, ) = wFd

f,ww iff(z, ) =

v,wFdfv,wz

vw.

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 ofS

Rj and U

Rj in the L

2(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, ) f(z, ) z1

j , (2.3)

UR[]j : f(z, ) f(0, ) j+ f(z, ) z1j . (2.4)

Note that

UR[]i S

Rj : f(z, ) UR[]i (f(z, ) zj) =i,jf(z, )

and hence we have the useful identity

UR[]i S

Rj =i,jI. (2.5)

On the other handSRj U

R[]j : f(z, ) SRj (f(0, )j + f(z, )z1j )

=f(0, )jzj+ [f(z, )z1j ]zj

and henceI dj=1

SRj UR[]j

: f(z, ) f(z, ) dj=1

f(0, )jzjdj=1

[f(z, )z1j ]zj

=f(0, ) dj=1

f(0, )jzj

and hence I dj=1

SRj UR[]j

: f(z, ) f(0, ) 1 d

j=1

zjj

. (2.6)Now suppose thatU= (U1, . . . ,Ud) is a row-unitaryd-tuple of operators on a

Hilbert space

K,

Eis a subspace of

K, and we define a map :

K L(

Fd

Fd,

E) by

k=

v,wFd(PEUwUvk)zvw. (2.7)


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Then

U

jk= v,wFd(PEUwUvUjk)zvw=wFd

(PEUwgjk)w +

v,w : v=(PEUwUvg

1j k)zvw

= (URj k)(z, ) (2.8)while

Uj k=

v,wFd(PEUvUwUj k)zvw

= v,wFd(PEUvUwgjk)zvw= (S

R[]j k)(z, ). (2.9)

If we let W = [] (where [] is the adjoint of with respect to the Hilbertspace inner product on K and the formalL2-inner product onL(FdFd, E)), then

[] : ze UUeandW:= [] = [Wv,w;,]v,w,,Fd where

Wv,w;, =PEUw

Uv

U

U

|E. (2.10)If im is given the lifted norm ,

k= P(ker)kK ,then one easily checks that W P(Fd Fd, E) im and

W p2 = []p2K= []p, []pK= Wp,pL2.

Thus, if we define a spaceLWas the closure ofW P(Fd Fd, E) in the normW p2LW = Wp,pL2 , (2.11)

thenLW = im isometrically. From the explicit form (2.10) ofWv,w;, it is easyto verify the intertwining relations

URj W =W SRj , S

R[]j W =W U

R[]j onP(Fd Fd, E).

If we defineUW = (UW,1, . . . ,UW,d) onLW byUW,j : W p URj W p= W Sjp, (2.12)

then, from the intertwining relations

Uj =URj , Uj =SR[]j forj = 1, . . . , d


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Functional Models 11

and the fact that is coisometric, we deduce thatUW is row-unitary onLW withadjointUJ = (UW,1, . . . ,UW,d) onLW given by

UW,j : W p SR[]j W p= W UR[]j p. (2.13)If is injective (or, equivalently, ifE is-cyclic for the row-unitary d-tupleU = (U1, . . . ,Ud)), thenUW = (UW,1, . . . ,UW,d) onLW is a functional modelrow-unitaryd-tuple for the abstractly given row-unitary d-tupleU= (U1, . . .Ud).

Our next goal is to understand more intrinsically which weights

W = [Wv,w;,]

can be realized in this way as (2.10) and thereby lead to functional models for

row-unitaryd-tuplesU. From identity (2.5) we see that (SR1, . . . , S Rd) becomes arow-isometry if we can change the inner product on L2(Fd Fd, E) so that theadjointSRj ofSj in the new inner product is U

R[]j . Moreover, if we in addition

arrange for the new inner product to have enough degeneracy to guarantee that

all elements of the form f(0, )(1 dj=1 zjj) have zero norm, then the d-tuple(SR1, . . . , S

Rd) 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-unitaryU.

LetWbe a function from four copies ofFd into bounded linear operators onEwith value at (v,w,,) denoted by Wv,w;,. We think ofWas a matrix withrows and columns indexed byFd Fd; thus Wv,w;, is the matrix entry for row(v, w) and column (, ). Denote byP(Fd, Fd, E) the space of all polynomials inthe non-commuting variables z1, . . . , zd, 1, . . . , d:

P(Fd, Fd, E) ={p(z, ) =

v,wFdpv,wz

vw : pv,w E and

pv,w = 0 for all but finitely many v, w}.ThenWcan be used to define an operator from P(FdFd, E) intoL(FdFd, E)by extending the formula

W: e z

v,wFdWv,w;,e z

vw.

for monomials to all ofP(Fd, Fd, E) by linearity. Note that computation of theL2-inner productWp,qL2 involves only finite sums ifp and qare polynomials,and therefore is well defined. We say that W is positive semidefinite if

Wp,pL2 0 for all p P(Fd Fd, E).

Under the assumption that Wis positive semidefinite, define an inner product onW P(Fd Fd, E) byW p , W q LW = Wp,qL2 . (2.14)


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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 ofW P(Fd Fd, E) in the inner product (2.14). (2.15)Note that the (v, w)-coefficient ofW p WP(Fd Fd, E) is given by

[W p]v,w, eE= Wp,W(zvwe)LW (2.16)and hence the map v,w : f fv,w extends continuously to the completionLWofW P(Fd Fd, E) andLW can be identified as a space of formal power seriesin the non-commuting variables z1, . . . , zd and 1, . . . , d, i.e., as a subspace ofL(FdFd, E). (This is the main advantage of defining the space as the completionofW

P(

Fd

Fd,

E) rather than simply as the completion of

P(

Fd

Fd,

E) in the

inner product given by the right-hand side of (2.14).) Note that then, for f LWand, FD we have

,f, eE = f, , eE= f, W[ze]LWfrom which we see that , :E LW is given by

, : e W[ze].By using this fact we see that

v,w,e, eE= ,e, v,weLW= W[ze], W[zvwe]LW= W[ze], zvweLW= W[ze], zvweL2= Wv,w;,e, eE

and we recover the operator matrix entries Wv,w;, of the operator W from thefamily of operators , (, Fd) via the factorization

Wv,w;, = v,w, .

Conversely, one can start with any such factorization of W (through a generalHilbert spaceKrather thanK = LWas in the construction above). The followingtheorem summarizes the situation.

Theorem 2.2. Assume that W = [Wv,w;, ]v,w,,Fd is a positive semidefinite(FdFd)(FdFd)matrix of operators on the Hilbert spaceEwith a factorizationof the form

Wv,w;, = v,w,

for operatorsv,w :K E for some intermediate Hilbert spaceK for allv, w, , F

d. Define an operator :K

L(Fd

Fd,

E) by

: k v,wFd

(v,wk)zvw.


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LetLWbe the Hilbert space defined as the completion of WP(Fd Fd, E) in thelifted inner product

W p , W pLW = Wp,pL2.Then is a coisometry fromK ontoLWwith adjoint given densely by

: W p

v,wFd,p,

forp(z, ) =

,p,z a polynomial inP(Fd Fd, E). In particular,LW is

given more explicitly asLW =im.Proof. Note thanW(as a densely defined operator onL2(FdFd, E) with domaincontaining at leastP(Fd Fd, E)) factors as W = [], where [] is the formalL2

-adjoint of defined at least on polynomials by[] : p(z, w)

,

,p, for p(z, w) =,

p,z .

In particular, []P(Fd Fd, E) is contained in the domain of when is con-sidered as an operator fromK intoLWwith dom ={k K : k LW}. Since is defined in terms of matrix entries v,w and evaluation of Fourier coefficientsp pv,w is continuous inLW, it follows that as an operator fromK intoLWwith domain as above is closed. For an element k K of the form []p for apolynomial p

P(

Fd

Fd,

E), we have

W p , W pLW = Wp,pL2= []k, kL2= []k, []kK.

Hence maps []P(Fd Fd, E) isometrically onto the dense submanifold WP(Fd Fd, E) ofLW. From the string of identities

k, []pK = k, pL2=

k , W p

LW

and the density of W P(Fd Fd, E) inLW, we see that ker = ([]P(FdFd, E)). Hence is isometric from a dense subset of the orthogonal complement ofits kernel ([]P(FdFd, E)) onto a dense subset ofLW (namely,WP(FdFd, E)).Since is closed, it follows that necessarily :K LW is a coisometry. Finally,notice that

k, W pLW = k, pL2= k, []pK

from which we see that

:W p []p for p P(Fd Fd, E)and the formula for follows. This completes the proof of Theorem 2.2.


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

UW,j: W p W SRj pand then extended to all ofLWby continuity become a row-isometry, or even arow-unitary operator-tuple. From (2.5), the row-isometry property follows if it can

be shown thatUW,j : W p W UR[]j p, or equivalently,

W SRj =URj W onP(Fd Fd, E). (2.17)

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

in addition that

W

p(0, )

1

dk=1

zkk

= 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. LetW be a(Fd Fd) (Fd Fd) matrix with matrix entries equalto operators on the Hilbert spaceE. Then:

1. W satisfies (2.17) if and only if

W,w;gj , =W,wgj;,, (2.19)

Wv,w;gj , =Wvg1j ,w;, forv= (2.20)

for allv ,w,, Fd, j = 1, . . . , d, where we interpret Wvg1j ,w;, to be 0in casevg1j is not defined.

2. Assume thatW is selfadjoint. ThenW satisfies (2.17)and (2.18)if and onlyifW satisfies (2.19), (2.20) and in addition

W,w;, =dj=1

W,wgj ;,gj (2.21)

for allw, Fd.

Proof. By linearity, it suffices to analyze the conditions (2.17) and (2.18) on mono-mialsf(z, ) =ze for some , Fd ande E. We compute

W SRj (ze) =W(zgje)

= v,wFd

Wv,w;gj ,zvw (2.22)


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and hencedj=1

Wv,w;gj ,gj =Wv,w;,gk

=Wvgk,w;, (by (2.24))=Wv,w;,

and hence (2.25) for this case is already a consequence of (2.20) and (2.24). Thiscompletes the proof of Theorem 2.3.

Note that (2.19) is a Hankel-like property for the operator matrix W for thisnon-commutative setting, while (2.20) is a Toeplitz-like property. We shall there-fore refer to operators W:

P(Fd Fd

,E

)

L(Fd Fd

,E

) for which both (2.19)and (2.20) are valid asHaplitz operators. We shall call positive semidefinite Haplitzoperators having the additional property (2.21) Cuntz weights. As an immediatecorollary of the previous result we note the following.

Corollary 2.4. LetWbe a positive semidefinite operator

W:P(Fd Fd, E) L(Fd Fd, E)and consider the d-tupleUW = (UW,1, . . . ,UW,d) of operators onLW defineddensely by

UW,j: W p

W SRj p forp

P(

Fd

Fd,

E). (2.26)

Then:

1. W is Haplitz if and only ifUW is a row-isometry.2. W is a Cuntz weight if and only ifUW is row-unitary.

In either case,UW,j andUW,j are then given on arbitrary elementsf LW byUW,j: f(z, ) f(0, ) 1j + f(z, ) zj , (2.27)UW,j : f(z, ) f(0, ) j+ f(z, ) z1j . (2.28)

Proof. The results are immediate consequences of (2.5) and (2.6). To prove (2.27)and (2.28), note the two expressions for

UW,j and for

UW,j

on polynomials in case

W =W is a Haplitz operator (as in (2.12) and (2.13)), and then note that thefirst formula necessarily extends by continuity to all ofLWsince the mapf fv,wis continuous on any spaceLW. Remark 2.5. If W = [Wv,w;,]v,w,,Fd where Wv,w;, = PEUwUvU

U |Efor a row-unitaryd-tupleUas in (2.10), then it is easily checked thatWis a Cuntzweight.To see this, note

W,w;gj , = PEUw UjUU

E

= PEUwgj

U

UE

= W,wgj ;,E


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simply from associativity of operator composition, and (2.19) follows. Similarly,for v= , we have

Wv,w;gj , = PEUwUv UjUU

E

= PEUwUvg1j UU

E

=Wvg1j ,w;,

from the row-isometry property ofU:

UiUj =i,jI.

Hence (2.20) follows andWis Haplitz. To check (2.21), we use the row-coisometry

property ofU (dj=1UjUj =I) to see thatdj=1

W,wgj ;,gj =dj=1

PEUwUjUjUE

= PEUwUE

=W,w;,.

From the formula (2.10) we see that Whas the additional normalization propertyW,;,=IE.

We shall be particularly interested in []-Haplitzoperators, i.e., (Fd Fd) (FdFd) operator matricesWfor which bothW andW[] are Haplitz. (Of coursea particular class of []-Haplitz operators are the selfadjoint Haplitz operators Haplitz operators W with W = W[].) For these operators the structure can bearranged in the following succinct way.

Proposition 2.6. Suppose thatW = [Wv,w;,]is a[

]-Haplitz operator matrix, and

define aFd Fd operator matrixW = [Wv,w] byWv,w =Wv,w;,.

ThenW is completely determined fromWaccording to the formulaWv,w;, =

W(v1),w if|v| ||,

W,w(v1) if|v| ||.(2.29)

Conversely, ifW is anyFd Fd matrix, then formula (2.29) defines a []-Haplitz operator matrixW.


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Proof. Suppose first that W = [Wv,w;,] is []-Haplitz. Then we compute, for|v| ||,

Wv,w;, =Wv1,w;, by (2.20)

=

W[],;v1,w

=

W[],;(v1),w

by (2.19) for W[]

=W(v1),w;,=W(v1),wwhile, for|v| || we have

Wv,w;, = W[],;v,w

=

W[]v1,;,w

by (2.20) for W[]

=W,w;v1,=W,w(v1);, by (2.19)

=

W[],;,w(v1)

=

W[],;,w(v1)

by (2.19)

=W

,w(v1

)

;,=W,w(v1)

and the first assertion follows.Conversely, givenW = [Wv,], define W = [Wv,w;,] by (2.29). Then verify

W,w;gj , =W,w versus W,wgj ;, =W,w and (2.19) follows for W.Similarly, compute, for v= ,

Wv,w;gj , =

W(vg1j 1),w if|v| > ||,W,w(gjv1) if|v| ||versus, again for v= ,

Wvg1j ,w;,=0 ifv=v gj,

Wv,w;, ifv=vgj

where

Wv,w;, =

Wv1,w if|v| ||,W,w(v1) if|v| ||and (2.20) follows for W.

From (2.29) we see that W[] is given by

W[]v,w;, = (W,;v,w)

=W(v1)w, if|| |v|,Ww,(v1) if|| |v|. (2.30)


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Using (2.30) we see that

W

[

]

,w;gj , = Wgjw,while

W[],wgj ;, =

Wgjw,and (2.19) follows for W[]. Similarly, for v= ,

W[]v,w;gj ,

=

Wgjv1)w, if|v| ||,

Ww,(vg1j 1)

if|v|> ||versus

W[]vg1j ,w;,

= 0 ifv=v gj,W

[]v,w;, ifv=v

gjwhere

W[]v,w;, =

W(v1)w, if|v| < ||,Ww,(v1) if|v| ||

and (2.20) follows for W[] as well. We conclude that W as defined by (2.29) is[]-Haplitz as asserted.

The formula (2.29) motivates the introduction of the symbolW(z, ) for the[]-Haplitz operatorW defined byW(z, )e= (W e)(z, ) = v,wFd

Wv,w;,ezvw.

For any []-HaplitzWand given any , Fd, we havev,w

Wv,w;,ezvw =W(ez)

=W((SR)

(UR[])

e)

= (U

R

)

W((U

R[

]

)

e)= (UR)

(SR[])

(W e).

Hence the matrix entries Wv,w;, for W are determined from a knowledge ofmatrix entries of the special form Wv,w;, (i.e., the Fourier coefficients of thesymbolW(z, )) via

v,wFd

Wv,w;,ezvw = (UR)

(SR[])

v,wFd

Wv,w;,ezvw

. (2.31)This gives a method to reconstruct a []-Haplitz operator directly from its symbol(equivalent to the reconstruction formula (2.29) from the matrix entries Wv,w;,).This can be made explicit as follows.


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The Haplitz operator W is a Cuntz weight if and only if its symbol

W(z, ) is a

positive symbol and its Cuntz defectD

W(z, ) is zero:

W(z, ) dj=1

z1jW(z, )1j = 0. (2.38)Proof. Suppose thatWis a positive semidefinite Haplitz operator. From the theoryof reproducing kernels, this means that Whas a factorization

Wv,w;, =Xv,wX, (2.39)

for some operatorsXv,w :L E. AsW is selfadjoint Haplitz, we have

Wv,w =Wv,w;,= W,;v,w=

W,v;,w

=W,w;,v

=X,wX,v

or Wv,w =YwYv (2.40)where we have set Yw =X,w. The identity (2.40) in turn is equivalent to (2.36)with Y(z) = wFd Ywzw.Derivation of the necessity of the factorization (2.37) lies deeper. From Corol-lary 2.4 we know that the operators (UW,1, . . . ,UW,d) given by (2.26) form a row-isometricd-tuple onLW. Furthermore, factorization (2.39) implies that the map

:

v,w,Fd(Xv,w)z

vw

is a coisometry. Without loss of generality, we may assume that is actuallyunitary. Hence there is a row-isometric d-tuple (V1, . . . , Vd) of operators onLdefined by

Vj =

UW,j for j = 1, . . . , d .

Set L = closed span{Y E: Fd}.We claim:Vj :Ye Ygje for e E andj = 1, . . . , d. Indeed, note that

Ye=v,w

(Xv,wYe)z

vw

=v,w

(Xv,wX,e)z

v, w

= v,w(Wv,w;,e)zvw= [W(e)](z, )


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from which we see that

UW,jY

e=

UW,jW(

e)

=W(UR[]j (

e))

=W(gje)

=v,w

(Wv,w;,gje)zvw

= Ygje

and the claim follows.Thus

L is invariant forVj for 1j d. As (V1, . . . , Vd) is a row-isometry,

it follows that (V1 , . . . , Vd )| L is a column contraction, ornj=1

Yjg1

...Yjgd

ej2

nj=1

Yjej2

for all 1, . . . , n Fd ande1, . . . , en E. Thus[Wvj ,vi ]i,j=1,...,n = [YviYvj ]i,j=1,...,n

Yv1

.

..Yvn

U1 . . . Ud U1

.

..Ud

Yv1 . . . Y vn=

d=1

Ygv1...Ygvn

Yv1g . . . Y vng

=d=1

[Wgvj ,vig ]i,j=1,...,n (2.41)and hence we have the inequalityWvj ,vi d

=1

Wgvj ,vigi,j=1,...,d

0. (2.42)

This is exactly the matrix form of the inequality (2.37), and hence (2.37) follows.Suppose now thatWis a Cuntz weight. Then (UW,1, . . . ,UW,d) is row-unitary.

It follows that (V1, . . . , Vd) is row-unitary, and hence that (V1 , . . . , Vd )| L is a col-umn isometry. It then follows that equality holds in (2.42), from which we get theequality (2.38) as asserted.

Conversely, suppose thatWis a selfadjoint Haplitz for which the two factor-izations (2.36) and (2.37) hold. Then we haveWv,w =YwYv where Yv :L E.


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We may assume that

L= closed span{Yv E: v Fd}. From (2.37), by reversing

the order of the steps in (2.41) we see that the operators (T1, . . . , Td) defined byTj Yve = Yvgje

extend to define a column contraction onL. By induction,TuYv = Yvue orYvTu =Yvu for all u Fd ande E, and

T1 . . . Td :L...

L

Lis a contraction. Let (

V1, . . . ,

Vd) on

L L be the row-isometric dilation of(T1, . . . , Td) (see [14]), soV1 . . . Vd :

L...L

L is isometric and PLVu = TuPL.Set

Xv,w =YwPLVv :L Efor v, w Fd. Then, for|v| || we have

Xv,wX, =YwPLVwV

Y=YwPLVv1Y=YwTv1Y=YwY

(v1)

=W(v1),w =Wv,w;,where we used (2.29) for the last step. Similarly, if|v| || we have

Xv,wX, =YwPLV

w

VY

=YwPLV(v1)Y=YwT(v1)Y=Yw(v1)Y

=W,w(v1)=Wv,w;,

and the factorization Wv,w, =Xv,wX, shows that W is positive semidefinite

as wanted.

If the symbolWof the Haplitz operator Wsatisfies (2.36) and (2.38), thenwe see that the d-tuple (T1,...,Td) in the above construction is a row-isometry, inwhich case the row-isometric extension (V1,...,Vd) of the row-coisometry (T1,...,Td)


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is in fact row-unitary. The construction above applies here to give Wv,w;, =YwP

LVvVY, but now withVequal to a row-unitary d-tuple. Hence

dj=1

W,wgj ;,gj =dj=1

YwgjPLYgj

=dj=1

YwPLVjVj Y=YwY

=W,w;,

and (2.21) follows, i.e.,Wis a Cuntz weight in case (2.37) is strengthened to (2.38).The theorem now follows.

Remark 2.9. IfWis a []-Haplitz operator, then the adjointW[] ofWhas symbolW[](z, ) = v,w

Wv,w;,zvw

where

Wv,w, = (W,;v,w)

= (W,v;,w)

=W[]

,w;

,v

=W[],;w,v

= (Ww,v;,)

from which we see that W[](z, ) =W(, z). (2.43)(where we use the convention that (zv) = zv

and (w) = w

). Thus theselfadjointness of a []-Haplitz operatorW can be expressed directly in terms ofthe symbol: W =W[] (as a []-Haplitz operator) if and only if

W(z, ) =W(, z). (2.44)Note that (2.44) is an immediate consequence of the factorization (2.36) requiredfor positive semidefiniteness of the Haplitz operator W.

In caseWis a Cuntz weight with the additional property

W,;,=IE, (2.45)

then the coefficient space Ecan be identified isometrically as a subspace ofLW viathe isometric map VW:E LW given by VW: eW e. Let us say that a CuntzweightWwith the additional property (2.45) is a normalized Cuntz weight. Then

we have the following converse of Corollary 2.4 for the case of normalized Cuntzweights. A more precise converse to Corollary 2.4, using the notion of row-unitaryscattering system, is given in [4].


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Theorem 2.10. LetU = (U1, . . . ,Ud) be a row-unitaryd-tuple of operators on theHilbert spaceK and letEbe a subspace ofK. LetW be the(Fd Fd) (Fd Fd)matrix of operators onE defined by

Wv,w;, = PEUwUvUUE

. (2.46)

ThenW is a normalized Cuntz weight and the map defined by

: k

v,wFd(PEUwUvk)zvw (2.47)

is a coisometry fromK ontoLW which satisfies the intertwining property

Uj =UW,j forj = 1, . . . , d .In particular, if the span of{UwUve : v, w Fd ande E} is dense inK, thenis unitary and the row-unitaryd-tupleUis unitarily equivalent to the normalizedCuntz-weight model row-unitaryd-tupleUW via . Moreover, ifW is any othernormalized Cuntz weight and any other coisometry fromK ontoLW such thatUj =UW,j forj = 1, . . . , dande= W(ze)for alle E, thenW =Wand = .

Proof. Apart from the uniqueness statement, Theorem 2.10 has already been de-rived in the motivational remarks at the beginning of this section and in Remark

2.5. We therefore consider only the uniqueness assertion.Suppose that W is another normalized Cuntz weight onE for which there

is a coisometry :K LW with : eWe for e E for all e Eand withUj =UW,j for j = 1, . . . , d.

We claim that W = W, or Wv,w;, = Wv,w;, for all v ,w,, Fd.Since W is also normalized, in fact is isometric onE, and hence also onspanv,wFdUwUvE. Hence

Wv,w;,e, eE = UwUvUUe, eK

= U

U

e,Uv

Uw

eK= (UUe), (UvUwe)LW= UWU

W e,UvWUw

W eLW

= UWU

W We,UvWUw

W WeLW

= W(ze), W(zvwe)LW= W(ze), zvweL2=

Wv,w;,e, e

E.

We conclude that W = W as claimed. This completes the proof of Theorem2.10.


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For purposes of the following corollary, let us say that two normalized CuntzweightsW andW on Hilbert spaces EandE respectively areunitarily equivalentif there is a unitary operator:E E such that Wv,w;, =Wv,w;,for allv,w,, Fd.Corollary 2.11. The assignment

(U1, . . . ,Ud; E) [Wv,w;, ] = [PEUwUvUU|E]of an equivalence class of normalized Cuntz weights to a row-unitary d-tuple(U1, . . . ,Ud) of operators on a Hilbert spaceK (or, equivalently, an isometric rep-resentation of the Cuntz algebraOd) together with a choice of-cyclic subspaceEprovides a complete unitary invariant for the category of row-unitaryd-tuples to-

gether with cyclic subspace. Specifically, if(U1, . . . ,Ud)is row-unitary on the HilbertspaceK with cyclic subspaceE K and (U1, . . . ,Ud) is another row-unitary onthe Hilbert spaceK with cyclic subspaceE, then there is a unitary transformation :K K withUj =Uj forj = 1, . . . , dand such thatE =E if and only ifWv,w;, =

Wv,w;, for allv ,w,, Fd, whereWv,w;, =PEUwUvUU|E,Wv,w;, =P

EUwUvU

U|E , andthe unitary:

E E is given by=

|E.

Remark 2.12. Theorem 2.10 can be formulated directly in terms of the modelrow-unitaryUW = (UW,1, . . . ,UW,d) on the model spaceLW as follows. ForW aCuntz weight onE, the matrix entriesWv,w;, (forv ,w,, Fd) are given by

Wv,w;, =iWUwWUvWU

WU

W iW (2.48)

where iW:E LW is the injection operator i : e W e. To see (2.48) directly,note that

Wv,w;,e, e

E=

W(ze), zvwe

L2

= W(ze), W(zvwe)LW= W SR(UR[])e,WSRv(UR[])weLW= UWU

W W e,Uv

WUw

W W eLW

= iWUwWUvWU

WU

W iWe, eEand (2.48) follows.

Remark 2.13. From the point of view of classification and model theory for rep-

resentations of the Cuntz algebra, the weakness in Corollary 2.11 is the demandthat a choice of cyclic subspaceE be specified. To remove this constraint whatis required is an understanding of when two Cuntz weights W and W are such


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that the corresponding Cuntz-algebra representationsUW andUW are unitarilyequivalent, i.e., when is there a unitary intertwining mapS:LW LW such that

SUW,j=UW,jS for j = 1, . . . , d .Preliminary results on this problem appear in Section 3 where the analyticintertwining maps S (S:HW HW ) are analyzed.Remark 2.14. An instructive exercise is to sort out the model for the case d= 1.Then the alphabet consists of a single letter g and the semigroupFd can beidentified with the semigroup Z+ of non-negative integers (with the empty word set equal to 0 and a word w = g . . . g set equal to its length|w| N Z+). Hencea Haplitz weightWis a matrix with rows and columns indexed byZ+ Z+. TheHaplitz property (2.19) means

W0,n;k+1, =W0,n+1;k, (2.49)

while (2.20) meansWm+1,n;k+1, =Wm,n;k,. (2.50)

Condition (2.21) reduces to

W0,n;0,k =W0,n+1;0,k+1. (2.51)

and (2.29) becomes

Wm,n;k, = Wmk+,n ifm k,W,n+km ifm k

where Wi,j =Wi,j;0,0 = (W0,0;i,j) = (W0,i;0,j)=W0,j;0,i=

W0,ji;0,0 =W0,ji ifj i,W0,0;0,ij ifj i

where, for j i,W0,0;0,ij = (W0,ij;0,0) = (

W0,ij)

or alternatively

W0,0;0,ij = (W0,ij;0,0) = (W0,0;ij,0) =Wij,0;0,0=Wij,0.Hence if we set

Tk =

W0,k ifk 0,Wk,0 = (W0,k) ifk 0,thenWi,j =Tij andWm,n;k, collapses to

Wm,n;k, =Tn+km.

The spaceLW is spanned by elements of the form W(zke) for k, Z+ ande E. However, from (2.51) we see that W(z

k

e) = W(zk+1

+1

e) and hencewe may identify W(zke) simply with W(zke) (where nowk in general liesin Z). The reduced weightWhas row and columns indexed by Z and is given by


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Wm,k =Tkm. In this way, we see that a Haplitz weight for the case d = 1 reducesto a Laurent matrix [Tkm]m,kZ. If we then solve the trigonometric moment

problem to produce a operator-valued measure on the unit circle such that

Tj =

T

zj(z),

we then obtain a version of the spectral theorem for the unitary operatorU.

3. Analytic intertwining operators between model spaces

Let W and W be two Cuntz weights (or more generally positive semidefiniteHaplitz operators) with block-matrix entries equal to operators on Hilbert spaces

E andE respectively. Let us say that a bounded operator S:LW LW is anintertwining operatorif

SUW,j=UW,jS, SUW,j=UW,jSforj = 1, . . . , d. WhileLWis the analogue of the Lebesgue space L2, the subspaceHW := closure inLW ofWP(Fd {}, E) is the analogue of the Hardy spaceH2. Let us say that an intertwining operator S:LW LW with the additionalpropertyS:HW HW is ananalytic intertwining operator. Note that for d= 1withLW andLW simply taken to be L2(Z, E) and L2(Z, E), the contractiveanalytic intertwining maps are multiplication operators with multiplier T fromthe Schur classS(E, E) of contractive, analytic, operator-valued functions on theunit disk. In analogy with this classical case, we denote bySnc(W, W) (the non-commutative Schur class associated with Cuntz weights W and W) the class ofall contractive, analytic, intertwining operators fromLW toLW . A particularlynice situation is when there is a formal power series T(z) so that

S(W e) =W[T(z)e] for each e E. (3.1)When we can associate a formal power series T(z) with the operator SSnc(W,W)in this way, we think of T(z) as the symbol for S and write

S(z) = T(z). The

purpose of this section is to work out some of the calculus for intertwining mapsS Snc(W, W) and their associated symbols T(z), and understand the conversedirection:given a power seriesT(z), when can the formula (3.1)be used to definean intertwining operator S Snc(W, W)?Two particular situations where thisoccurs are: (i) W is a general positive Haplitz operator and T(z) is an analyticpolynomial (so we are guaranteed thatT(z)eis in the domain ofW), and (ii)Wis a Haplitz extension of the identity and T(z)e L2(Fd, E) for each e E (soagainT(z)eis guaranteed to be in the domain ofW). Here we say that the HaplitzoperatorW is aHaplitz extension of the identityifW;v,;,=v,IE wherev,is the Kronecker delta-function equal to 1 forv = and equal to 0 otherwise. The

setting (ii) plays a prominent role in the analysis of Cuntz scattering systems andmodel and dilation theory for row-contractions in [3]. Our main focus here is onthe setting (i).


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In general, given a formal power series T(z) =

wFd Twz

w in the non-commuting variables z = (z1, . . . , zd) with coefficients Tw equal to bounded oper-

operator theory systems theory and scattering theory multidimensional generalizations

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