Operator Theory: Advances and
Applications
Vol. 157
Editor:
I. Gohberg
Editorial Office:
School of Mathematical
Sciences
Tel Aviv University
Ramat 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)
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 Advisory
Editorial Board:
C. Foias (Bloomington)
P. R. Halmos (Santa Clara)
T. Kailath (Stanford)
P. D. Lax (New York)
M. S. Livsic (Beer Sheva)
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Operator Theory, Systems Theory
and Scattering Theory:
Multidimensional Generalizations
Daniel Alpay
Victor Vinnikov
Editors
Birkhäuser Verlag
Basel . Boston . Berlin
www.pdfgrip.com
Editors:
Daniel Alpay
Victor Vinnikov
Department of Mathematics
Ben-Gurion University of the Negev
P.O. Box 653
Beer Sheva 84105
Israel
e-mail:
2000 Mathematics Subject Classification 47A13, 47A40, 93B28
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
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ISBN 3-7643-7212-5 Birkhäuser Verlag, Basel – Boston – Berlin
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Contents
Editorial Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
J. Ball and V. Vinnikov
Functional Models for Representations of the Cuntz Algebra . . . . . . . . .
1
T. Banks, T. Constantinescu and J.L. Johnson
Relations on Non-commutative Variables and
Associated Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
M. Bessmertny˘ı
Functions of Several Variables in the Theory of Finite
Linear Structures. Part I: Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
S. Eidelman and Y. Krasnov
Operator Methods for Solutions of PDE’s
Based on Their Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
D.S. Kalyuzhny˘ı-Verbovetzki˘ı
On the Bessmertny˘ı Class of Homogeneous Positive
Holomorphic Functions on a Product of Matrix Halfplanes . . . . . . . . . . . 139
V. Katsnelson and D. Volok
Rational Solutions of the Schlesinger System and
Isoprincipal Deformations of Rational Matrix Functions II . . . . . . . . . . .
165
M.E. Luna–Elizarrar´
as and M. Shapiro
Preservation of the Norms of Linear Operators Acting
on some Quaternionic Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
P. Muhly and B. Solel
Hardy Algebras Associated with W ∗ -correspondences
(Point Evaluation and Schur Class Functions) . . . . . . . . . . . . . . . . . . . . . . . . 221
M. Putinar
Notes on Generalized Lemniscates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
243
M. Reurings and L. Rodman
One-sided Tangential Interpolation for Hilbert–Schmidt
Operator Functions with Symmetries on the Bidisk . . . . . . . . . . . . . . . . . .
267
F.H. Szafraniec
Favard’s Theorem Modulo an Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
301
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Operator Theory:
Advances and Applications, Vol. 157, viixvi
c 2005 Birkhă
auser Verlag Basel/Switzerland
Editorial Introduction
Daniel Alpay and Victor Vinnikov
La s´eduction de certains probl`emes vient de leur
d´efaut de rigueur, comme des opinions discordantes qu’ils suscitent: autant de difficult´es 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 theory
in the multi-dimensional case. This last term includes a wide range of situations
and we review the one variable case first.
An important player in the single variable theory is a contractive analytic function on the open unit disk. Such functions, often called Schur functions, have a
rich theory of their own, especially in connection with the classical interpolation
problems. They also have different facets arising from their appearance in different
areas, in particular as:
• characteristic operator functions, in operator model theory. Pioneering works
include the works of Livˇsic and his collaborators [54], [55], [25], of Sz. Nagy
and Foia¸s [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 solution
of the latter using reproducing kernel Hilbert space methods, see [8], [9].
• transfer functions, in system theory. It follows from the Bochner–Chandrasekharan theorem that a system is linear, time-invariant, and dissipative if
and only if it has a transfer function which is a Schur function. For more
general systems (even multi-dimensional ones) one can make use of Schwartz’
kernel theorem (see [76], [52]) to get the characterisation of invariance under
translation; 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 only
one underlying theory.
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viii
D. Alpay and V. Vinnikov
One natural extension of the single variable theory is the time varying case, where
one (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 various
approaches of the standard one variable theory generalize together with their interrelations. On the other hand, in the multi-dimensional case there is no longer
a single underlying theory, but rather different theories, some of them loosely
connected and some not connected at all. In fact, depending on which facet of
the one-dimensional case we want to generalize we are led to completely different
objects and borderlines between the various theories are sometimes vague. The
directions 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 system theory, one is dealing here with the conservative version of the systems
known as the Roesser model or the Fornasini–Marchesini model in the multidimensional 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 problem
for formal power series in non-commuting variables that appeared rst in
the theory of automata, see Schă
utzenberger [74], [75] and Fliess [44], [45]
(for a good survey see [17]), and more recently in robust control of linear
systems subjected to structured possibly time-varying uncertainty (see Beck,
Doyle and Glover [15] and Lu, Zhou and Doyle [59]). In operator theory, two
main parallel directions may be distinguished; the first direction is along the
lines of the works of Drury [43], Frazho [47], [48], Bunce [26], and especially
the vast work of Popescu [65], [63], [64], [66], where various one-dimensional
models are extended to the case of several non-commuting operators. Another
direction is related to the representations of the Cuntz algebra and is along
the line of the works of Davidson and Pitts (see [36] and [37]) and Bratelli
and 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 Nevanlinna–Pick
kernels; see the works of Quiggin [70], McCullough and Trent [60] and Agler
and McCarthy [3]. We note also connections with the theory of wavelets and
with system theory on trees; see [16], [10].
• Hyponormal operators, subnormal operators, and related topics. Though nominally dealing with a single operator, the theory of hyponormal operators and
of certain classes of subnormal operators has many features in common with
multivariable operator theory. We have in mind, in particular, the works of
Putinar [68], Xia [81], and Yakubovich [82]. For an excellent general survey of
the theory of hyponormal operators, see [80]. Closely related is the principal
function theory of Carey and Pincus, which is a far reaching development
of the theory of Kre˘ın’s spectral shift function; see [62], [27], [28]. Another
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Editorial Introduction
ix
closely related topic is the study of multi-dimensional moment problems; of
the 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) 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 difficulty
is that the product of two (say, left) hyperanalytic functions need not be
left hyperanalytic. Setting the real part of the quaternionic variable to be
zero, one obtains a real analytic quaternionic-valued function. Conversely,
the Cauchy–Kovalevskaya theorem allows to associate (at least locally) to
any such function a hyperanalytic function. Identifying the quaternions with
C2 one obtains an extension of the theory of functions of one complex variable
to maps from (open subsets of) C2 into C2 . Rather than two variables there
are now three non-commutative non-independent hyperanalytic variables and
the counterparts of the polynomials z1n1 z2n2 are now non-commutative polynomials (called the Fueter polynomials) in these hyperanalytic variables. The
original 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 Schlesinger [73] and Garnier [51], is to represent the non-linear equation as the
compatibility condition for some over-determined linear differential system
and consider the corresponding families (so-called deformations) of ordinary
linear equations. From the view point of this theory, the situation when the
linear equations admit rational solutions is exceptional: the non-resonance
conditions, the importance of which can be illustrated by Bolibruch’s counterexample to Hilbert’s 21st problem (see [11]), are not met. However, analysis of this situation in terms of the system realization theory may lead to
explicit 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˘ı, which foreshadowed many subsequent developments and contains
a 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 interpolation
for Hilbert–Schmidt operator functions with symmetries on the bidisk, deals with
interpolation in the bidisk in the setting of H 2 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 authors
develop functional models and a certain theory of Fourier representation for a representation of the Cuntz algebra (i.e., a row unitary operator). Next we have the
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D. Alpay and V. Vinnikov
paper of Banks, Constantinescu and Johnson, Relations on non-commutative variables and associated orthogonal polynomials, where the authors survey various
settings where analogs of classical ideas concerning orthogonal polynomials and
associated positive kernels occur. The paper serves as a useful invitation and orientation for the reader to explore any particular topic more deeply. In the paper
of Kalyuzhny˘ı-Verbovetzki˘ı, On the Bessmertny˘ı class of homogeneous positive
holomorphic functions on a product of matrix halfplanes, a recent investigation
of the author on the Bessmertny˘ı class of operator-valued functions on the open
right poly-halfplane which admit a so-called long resolvent representation (i.e., a
Schur complement formula applied to a linear homogeneous pencil of operators
with positive semidefinite operator coefficients), is generalized to a more general
“non-commutative” domain, a product of matrix halfplanes. The study of the Bessmertny˘ı class (as well as its generalization) is motivated by the electrical networks
theory: as shown by M.F. Bessmertny˘ı [18], for the case of matrix-valued functions for which finite-dimensional long resolvent representations exist, this class
is exactly the class of characteristic functions of passive electrical 2n-poles where
impedances of the elements of a circuit are considered as independent variables.
Finally, in the paper Hardy algebras associated with W ∗ -correspondences (point
evaluation and Schur class functions), Muhly and Solel deal with an extension of
the non-commutative theory from the point of view of non-self-adjoint operator
algebras.
Hyponormal and subnormal operators and related topics:
The paper of Putinar, Notes on generalized lemniscates, is a survey of the theory
of domains bounded by a level set of the matrix resolvent localized at a cyclic
vector. The subject has its roots in the theory of hyponormal operators on the one
hand and in the theory of quadrature domains on the other. While both topics are
mentioned 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 an
algebraic set (or more generally with respect to a positive semidefinite functional)
and three term recurrence relations. As it happens often the general approach
sheds new light also on the classical one-dimensional situation.
Hyperanalytic functions:
In the paper Operator methods for solutions of differential equations based on
their symmetries, Eidelman and Krasnov deal with construction of explicit solutions for some classes of partial differential equations of importance in physics, such
as evolution equations, homogeneous linear equations with constant coefficients,
and analytic systems of partial differential equations. The method used involves
an explicit construction of the symmetry operators for the given partial differential 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 symmetry operators on the “simplest” solution. This allows to obtain representations
of Clifford-analytic functions in terms of power series in operator indeterminates.
Luna–Elizarrar´
as and Shapiro in Preservation of the norms of linear operators acting on some quaternionic function spaces consider quaternionic analogs of some
classical real spaces and in particular compare the norms of operators in the original 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 and
Volok, Rational solutions of the Schlesinger system and rational matrix functions
II, which presents an explicit construction of the multi-parametric holomorphic
families of rational matrix functions, corresponding to rational solutions of the
Schlesinger non-linear system of partial differential equations.
There are many other directions that are not represented in this volume. Without
the pretense of even trying to be comprehensive we mention in particular:
• Model theory for commuting operator tuples subject to various higher-order
contractivity 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 the
joint spectrum [78] and of the analytic functional calculus [77] for commuting
operators (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, with
applications 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 main
notion is that of an operator vessel, due to Livˇsic; see [56], [57], [58]. This
turns 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 variables; 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 at
the Department of Mathematics of Ben-Gurion University of the Negev during the
period June 30–July 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 Mathematics of Ben-Gurion University of the Negev for its generosity and for making the
workshop possible.
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Daniel Alpay and Victor Vinnikov
Department of Mathematics
Ben-Gurion University of the Negev
Beer-Sheva, Israel
e-mail:
e-mail:
www.pdfgrip.com
Operator Theory:
Advances and Applications, Vol. 157, 160
c 2005 Birkhă
auser Verlag Basel/Switzerland
Functional Models for Representations
of the Cuntz Algebra
Joseph A. Ball and Victor Vinnikov
Abstract. We present a functional model, the elements of which are formal
power series in a pair of d-tuples of non-commuting variables, for a row-unitary
d-tuple of operators on a Hilbert space. The model is determined by a weighting matrix (called a “Haplitz” matrix) which has both non-commutative Hankel and Toeplitz structure. Such positive-definite Haplitz matrices then serve
to classify representations of the Cuntz algebra Od with specified cyclic subspace up to unitary equivalence. As an illustration, we compute the weighting
matrix for the free atomic representations studied by Davidson and Pitts and
the 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=−∞ fn z n
by
∞
Φ: k →
(PE U ∗n k)z n
n=−∞
where PE is the orthogonal projection onto the subspace E ⊂ K. Note that Φ(k) = 0
if and only if k is orthogonal to the smallest reducing subspace for U containing
the 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 norm
∞
∞
2
< ∞}.
square-summable series L2 (T, E) = {f (z) = n=−∞ fn z n :
n=−∞ fn
The first author is supported by NSF grant DMS-9987636; both authors are support by a grant
from the US-Israel Binational Science Foundation.
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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
2
L
2
K.
= P(ker Φ)⊥ k
∞
n=−∞
Moreover, we see that if we set Φk =
then
∞
fn z n for a k ∈ K (so fn = PE U ∗n k),
(PE U ∗n−1 k)z n
ΦUk =
n=−∞
∞
fn−1 z n
=
n=−∞
∞
=z·
fn−1 z n−1
n=−∞
∞
=z·
fn z n = Mz Φk,
n=−∞
i.e., the operator U is now represented by the operator Mz of multiplication by
the variable z on the space L.
We can make this representation more explicit as follows. The standard adjoint Φ[∗] of Φ with respect to the L2 -inner product on the target domain is defined
at least on polynomials:
⎛
⎞
N
pj z j
Φk,
j=−N
where we have set
Φ[∗] ⎝
N
pj z j ⎠
j=−N
L2
⎛
N
k, Φ[∗] ⎝
=
⎞
pj z j ⎠ =
j=−N
K
N
U j pj .
j=−N
Furthermore, the range Φ[∗] P of Φ[∗] acting on polynomials (where we use P
to denote the subspace of L2 (T, E) consisting of trigonometric polynomials with
coefficients 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
[∗]
= ΦΦ p, p
K
L2 .
This suggests that we set W = ΦΦ[∗] (well defined as an operator from the space
of E-valued polynomials P to the space L(Z, E) of formal Fourier series with coefficients in E) and define a Hilbert space LW as the closure of W P in the inner
product
W p, W q LW = W p, q L2 .
The Toeplitz structure of W (i.e., the fact that Wi,j = PE U j−i |E depends only on
the difference i − j of the indices) implies that the operator Mz of multiplication
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Functional Models
3
by z is isometric (and in fact unitary) on LW . Conversely, starting with a positive
semidefinite 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 acting
on LW as a functional model for a unitary operator. While the space LW in
general consists only of formal Fourier series and there may be no bounded point
evaluations for the elements of the space, evaluation of any one of the Fourier
coefficients is a bounded operator on the space, and gives the space at least the
structure of a formal reproducing kernel Hilbert space, an L2 -version of the usual
reproducing kernel Hilbert spaces of analytic functions arising in many contexts;
we develop this idea of formal reproducing kernel Hilbert spaces more fully in the
separate report [4].
Note that a unitary operator can be identified with a unitary representation
of the circle group T or of the C ∗ -algebra C(T). Given any group G or C ∗ -algebra
A, there are two natural problems: (1) classification up to unitary equivalence of
unitary representations of G or of A, and (2) classification up to unitary equivalence
of unitary representations which include the specification of a ∗-cyclic subspace.
While the solution of the first problem is the loftier goal, the second problem is
arguably 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 not
the case, a solution of the second problem often can be used as a stepping stone
to a solution of the first problem. In the case of G = T or A = C(T), the theory of
LW spaces solves the second problem completely: given two unitary operators U on
K and U on K with common cyclic subspace E contained in both K and K , then
there is a unitary operator U : K → K satisfying U U = U U and U |E = IE if and
only if the associated Toeplitz matrices Wi,j = PE U j−i |E and Wi,j = PE U j−i |E
are identical, and then both U and U are unitarily equivalent to UW on LW with
canonical cyclic subspace W · E ⊂ LW . A little more work must be done to analyze
the dependence on the choice of cyclic subspace E and thereby solve the first
classification problem. Indeed, if we next solve the trigonometric moment problem
for W and find a measure µ on T (with values equal to operators on E) for which
Wn = T z n dµ(z), then we arrive at a representation for the original operator U as
the multiplication operator Mz on the space L2 (µ). Alternatively, one can use the
theory of the Hellinger integral (see [5]) to make sense of the space of boundary
values 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 Hilbert
space L(ϕ) appearing prominently in work of de Branges and Rovnyak in their
approach to the spectral theory for unitary operators (see, e.g., [6]), where
ϕ(z) =
T
λ+z
dµ(z) for z in the unit disk D,
λ−z
under the transformation (f (z), g(z)) → f (z) + z −1 g(z −1 ). In any case, the first
(harder) classification problem (classification of unitary representations up to unitary equivalence without specification of a ∗-cyclic subspace) is solved via use of
the equivalence relation of mutual absolute continuity on spectral measures. For
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4
J.A. Ball and V. Vinnikov
this classical case, we see that the solution of the second problem serves as a
stepping 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 the
trigonometric 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, a
d-tuple of operators U = (U1 , . . . , Ud ) on a Hilbert space K which is row-unitary,
i.e.,
⎡ ∗⎤
⎡
⎤
⎡ ∗⎤
U1
I
U1
⎢ .. ⎥
⎥
⎢
⎥
⎢
.
..
U1 . . . Ud ⎣ ... ⎦ = I.
⎣ . ⎦ U1 . . . Ud = ⎣
⎦,
Ud∗
Ud∗
I
Equivalently, U = (U1 , . . . , Ud ) is a d-tuple of isometries on K with orthogonal
ranges and with span of the ranges equal to the whole space K. It is known that
Od is NGCR, and hence the first classification problem for the case of Od is intractable in a precise sense, although particular special cases have been worked
out (see [7, 9]). The main contribution of the present paper is that there is a satisfactory solution of the second classification problem (classification up to unitary
equivalence of unitary representations with specification of ∗-cyclic subspace) for
the case of Od via a natural multivariable analogue of the spaces LW sketched
above 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 of
the semigroup are words w of the form w = gin . . . gi1 with i1 , . . . , in ∈ {1, . . . , d}.
If w = gin . . . gi1 , set U w = Uin · · · Ui1 . The functional model for such a row-unitary
d-tuple will consist of formal power series of the form
fv,w z v ζ w
f (z, ζ) =
(1.1)
v,w∈Fd
where z = (z1 , . . . , zd ) and ζ = (ζ1 , . . . , ζd ) is a pair of d non-commuting variables.
The formalism is such that zi zj = zj zi and ζi ζj = ζj ζi for i = j but zi ζj = ζj zi for
all i, j = 1, . . . , d. In the expression (1.1), for w = gin · · · gi1 we set z w = zin · · · zi1
and similarly for ζ. The space LW of non-commuting formal power series which
serves as the functional model for the row-unitary U = (U1 , . . . , Ud ) with cyclic
subspace E will be determined by a weighting matrix
Wv,w;α,β = PE U w U ∗v U α U ∗β |E
with 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 have
the subspace W · E as a ∗-cyclic subspace. Matrices W (with rows and columns
indexed by Fd × Fd ) arising in this way from a row-unitary U can be characterized
by a non-commutative analogue of the Toeplitz property which involves both a
non-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”. Such
Cuntz weights serve as a complete unitary invariant for the second classification
problem for the Cuntz algebra Od : given two row-unitary d-tuples U = (U1 , . . . , Ud )
on K and U = (U1 , . . . , Ud ) on K with common ∗-cyclic subspace E contained in
both K and K , then there is a unitary operator U : K → K such that U Uj = Uj U
and U Uj∗ = Uj∗ U for j = 1, . . . , d and U |E = IE if and only if the associated Cuntz
weights Wv,w;α,β = PE U w U ∗v U α U ∗β |E and Wv,w;α,β = PE U w U ∗v U α U ∗β |E
are identical, and then both U and U are unitarily equivalent to the model rowunitary d-tuple UW = (UW,1 , . . . , UW,d ) acting on the model space LW with canonical ∗-cyclic subspace W · E ⊂ LW .
The parallel with the commutative case can be made more striking by viewing LW as a non-commutative formal reproducing kernel Hilbert space, a natural
generalization of classical reproducing kernel Hilbert spaces to the setting where
the elements of the space are formal power series in a collection of non-commuting
indeterminates; we treat this aspect in the separate report [4].
A second contribution of this paper is the application of this functional model
for row-unitary d-tuples to the free atomic representations and permutative representations of Od appearing in [9] and [7] respectively. These representations are
of two types: the orbit-eventually-periodic type, indexed by a triple (x, y, λ) where
x and y are words in Fd and λ is a complex number of modulus 1, and the orbitnon-periodic case, indexed by an infinite word x = gk1 gk2 · · · gkn · · · . Davidson and
Pitts [9] have identified which pairs of parameters (x, y, λ) or x give rise to unitarily
equivalent representations of Od , which parameters correspond to irreducible representations, and how a given representation can be decomposed as a direct sum or
direct integral of irreducible representations. The contribution here is to recover
these results (apart from the identification of irreducible representations) as an
application of the model theory of LW spaces and the calculus of Cuntz weights.
The approach shares the advantages and disadvantages of the de Branges-Rovnyak
model theory for single operators (see [6]). Once Cuntz weights W are calculated,
identifying unitary equivalences is relatively straightforward and obtaining decompositions 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 in
specific cases of interest. While these results are obtained in an elementary way
in [9], our results here show that a model theory calculus, a non-commutative
multivariable 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, Section
2 lays out the functional models for row-isometries and row-unitary operatortuples in particular. We show there that the appropriate analogue for a bi-infinite
Toeplitz matrix is what we call a “Haplitz operator”. Just as Toeplitz operators
∞
W = [Wi−j ]i,j=...,−1,0,1,... have symbols W (z) = n=−∞ Wn z n , it is shown that
associated with any Haplitz operator W is its symbol W (z, ζ), a formal power series
in two sets of non-commuting variables (z1 , . . . , zd ) and ζ1 , . . . , ζd ). These symbols
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6
J.A. Ball and V. Vinnikov
serve as the set of free parameters for the class of Haplitz operators; many questions
concerning a Haplitz operator W can be reduced to easier questions concerning
its symbol W (z, ζ). In particular, positivity of the Haplitz operator W is shown to
be equivalent to a factorization property for its symbol W (z, ζ) and for the Cuntz
defect DW (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∗
between two row-unitary model spaces LW and LW∗ , and
tertwining maps LW,W
T
the subclass of such maps (“analytic intertwining operators”) which preserve the
subspaces analogous to Hardy subspaces. The contractive, analytic intertwining
operators 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 models
are unitarily equivalent, or when a given functional model decomposes as a direct
sum or direct integral of internal pieces (modulo overlapping spaces). Section 4
gives the application of the model theory and calculus of Cuntz weights to free
atomic 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 (especially the material in Section 3) to study non-commutative analogues of LaxPhillips scattering and unitary colligations, how they relate to each other, and
how they relate to the model theory for row-contractions developed in the work
of 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 generic
element 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 belonging
to the alphabet {g1 , . . . , gd }. We shall write |w| for the length n of the word
w = αn · · · α1 . If w = αn · · · α1 and v = βm · · · β1 are words, then the product vw
of v and w is the new word formed by the juxtaposition of v and w:
vw = βm · · · β1 αn · · · α1 .
We define the transpose w of the word w = gin · · · gi1 by w = gi1 · · · gin . We
denote the unit element of Fd by ∅ (corresponding to the empty word). In particular, if gk is a word of unit length, we write gk w for gk αn · · · α1 if w = αn · · · α1 .
Although Fd is a semigroup, we will on occasion work with expressions involving
inverses of words in Fd ; the meaning is as follows: if w and v are words in Fd , the
expression wv −1 means w if there is a w ∈ Fd for which w = w v; otherwise we
say that wv −1 is undefined. An analogous interpretation applies for expressions
of the form w−1 v. This convention requires some care as associativity can fail: in
general it is not the case that (wv −1 ) · w = w · (v −1 w ).
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Functional Models
7
For E an auxiliary Hilbert space, we denote by (Fd , E) the set of all E-valued
functions v → f (v) on Fd . We will write 2 (Fd , E) for the Hilbert space consisting
of all elements f in (Fd , E) for which
f
2
2 (F
d ,E)
:=
f (v)
2
E
< ∞.
v∈Fd
Note that the space 2 (Fd , E) amounts to a coordinate-dependent view of the
Fock space studied in [1, 9, 10, 12, 13]. It will be convenient to introduce the
non-commutative Z-transform f → f (z) on (Fd , E) given by
f (w)z w
f (z) =
v∈Fd
where z = (z1 , . . . , zd ) is to be thought of as a d-tuple of non-commuting variables,
and we write
z w = 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 S1R , . . . , SdR on 2 (Fd , E)
are given by
SjR : f → f where f (w) = f (wgj−1 )
with adjoint given by
SjR∗ : f → f where f (w) = f (wgj ).
(Here f (wgj−1 ) is interpreted to be equal to 0 if wgj−1 is undefined.) In the noncommutative frequency domain, these right creation operators (still denoted by
S1R , . . . , SdR for convenience) become right multiplication operators:
SjR : f (z) → f (z) · zj ,
SjR∗ : f (z) → f (z) · zj−1 .
In the latter expression z w · zj−1 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 operators
is often easier to handle; hence in the sequel we will work primarily in the noncommutative frequency-domain setting L(Fd , E) rather than in the time-domain
setting (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 : ⊕dk=1 K → K
is an isometry. Equivalently, each of U1 , . . . , Ud is an isometry on K and the image
spaces im U1 , . . . , im Ud are pairwise orthogonal. There are two extreme cases of
row-isometries U: (1) the case where U is row-unitary, i.e., U1 . . . Ud is unitary, or equivalently, im U1 , . . . , im Ud span the whole space K, and (2) the case
where U is a row-shift, i.e.,
span{im U v : |v| = n} = {0};
n≥0
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8
J.A. Ball and V. Vinnikov
here we use the non-commutative multivariable operator notation
U v = Uin . . . Ui1 if v = gin · · · gi1 .
A general row-isometry is simply the direct sum of these extreme cases by the
Wold decomposition for row-isometries due to Popescu (see [14]). It is well known
that the operators S1R , . . . , SdR provide a model for any row-shift, as summarized
in the following.
Proposition 2.1. The d-tuple of operators (S1R , . . . , SdR ) on the space L2 (Fd , E) is
a row-shift. Moreover, if U = (U1 , . . . , Ud ) is any row-shift on a space K, then U
⊕dk=1 Uk K .
is unitarily equivalent to (S1R , . . . , SdR ) on L2 (Fd , E), with E = K
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 (F
d ×Fd ,E)
:=
f (v, w)
2
< ∞.
v,w∈Fd
The Z-transform f → f for elements of this type is given by
f (v, w)z v ζ w .
f (z, ζ) =
v,w
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 that
each 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 see
that the weighted modifications of 2 (Fd ×Fd , E) which we shall introduce below do
collapse to 2 (Z, E) for the case d = 1. Similarly, one should think of L2 (Fd , E) as a
non-commutative version of the Hardy space H 2 (D, E) over the unit disk D, and of
the modifications of L2 (Fd × Fd, E) to be introduced below as a non-commutative
analogue of the Lebesgue space L2 (T, E) of measurable norm-square-integrable
E-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 convenient
to use non-commutative multiplication of formal power series; for this reason we
shall write simply f (z, ζ) for elements of the space rather than f (z, ζ). Unlike the
unilateral setting L2 (Fd , E) discussed above, there are two types of shift operators
on L2 (Fd × Fd , E) of interest, namely:
SjR : f (z, ζ) → f (z, ζ) · zj ,
UjR :
f (z, ζ) → f (0, ζ) ·
ζj−1
(2.1)
+ f (z, ζ) · zj
www.pdfgrip.com
(2.2)
Functional Models
9
where f (0, ζ) is the formal power series in ζ = (ζ1 , . . . , ζd ) obtained by formally
setting z = 0 in the formal power series for f (z, ζ):
f∅,w ζ w if f (z, ζ) =
f (0, ζ) =
w∈Fd
fv,w z v ζ w .
v,w∈Fd
One can think of SjR as a non-commutative version of a unilateral shift (even
in this bilateral setting), while UjR is some kind of bilateral shift. We denote by
and Uj
the adjoints of SjR and UjR in the L2 (Fd × Fd , E)-inner product
Sj
(to avoid confusion with the adjoint with respect to a weighted inner product to
appear below). An easy computation shows that
R[∗]
R[∗]
R[∗]
: f (z, ζ) → f (z, ζ) · zj−1 ,
(2.3)
R[∗]
: f (z, ζ) → f (0, ζ) · ζj + f (z, ζ) · zj−1 .
(2.4)
Sj
Uj
Note that
R[∗]
Ui
R[∗]
SjR : f (z, ζ) → Ui
(f (z, ζ) · zj ) = δi,j f (z, ζ)
and hence we have the useful identity
R[∗]
Ui
SjR = δi,j I.
(2.5)
On the other hand
R[∗]
SjR Uj
: f (z, ζ) →SjR (f (0, ζ)ζj + f (z, ζ)zj−1 )
= f (0, ζ)ζj zj + [f (z, ζ)zj−1 ]zj
and hence
⎛
⎝I −
⎞
d
R[∗] ⎠
SjR Uj
d
d
: f (z, ζ) →f (z, ζ) −
j=1
[f (z, ζ)zj−1 ]zj
f (0, ζ)ζj zj −
j=1
j=1
d
= f (0, ζ) −
f (0, ζ)ζj zj
j=1
and hence
⎛
⎝I −
d
⎞
R[∗] ⎠
SjR Uj
⎛
: f (z, ζ) → f (0, ζ) · ⎝1 −
j=1
d
⎞
zj ζj ⎠ .
(2.6)
j=1
Now suppose that U = (U1 , . . . , Ud ) is a row-unitary d-tuple of operators on a
Hilbert space K, E is a subspace of K, and we define a map Φ : K → L(Fd ×Fd , E) by
(PE U w U ∗v k)z v ζ w .
Φk =
v,w∈Fd
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(2.7)
10
J.A. Ball and V. Vinnikov
Then
(PE U w U ∗v Uj k)z v ζ w
Φ ◦ Uj k =
v,w∈Fd
v,w : v=∅
w∈Fd
=
−1
(PE U w U ∗vgj k)z v ζ w
(PE U wgj k)ζ w +
=
(UjR
◦ Φk)(z, ζ)
(2.8)
while
Φ ◦ Uj∗ k =
(PE U v U ∗w Uj∗ k)z v ζ w
v,w∈Fd
(PE U v U ∗wgj k)z v ζ w
=
v,w∈Fd
R[∗]
= (Sj
◦ Φk)(z, ζ).
(2.9)
If we let W = ΦΦ[∗] (where Φ[∗] is the adjoint of Φ with respect to the Hilbert
space inner product on K and the formal L2 -inner product on L(Fd × Fd , E)), then
Φ[∗] : z α ζ β e → U α U ∗β e
and W := ΦΦ[∗] = [Wv,w;α,β ]v,w,α,β∈Fd where
Wv,w;α,β = PE U w U ∗v U α U β |E .
·
If im Φ is given the lifted norm
Φk
(2.10)
,
= P(ker Φ)⊥ k
K
,
then one easily checks that W · P(Fd × Fd , E) ⊂ im Φ and
2
Wp
= Φ[∗] p
2
K
= Φ[∗] p, Φ[∗] p
= W p, p
K
L2 .
Thus, if we define a space LW as the closure of W · P(Fd × Fd , E) in the norm
Wp
2
LW
= W p, p
L2 ,
(2.11)
then LW = im Φ isometrically. From the explicit form (2.10) of Wv,w;α,β it is easy
to verify the intertwining relations
UjR W = W SjR ,
R[∗]
Sj
R[∗]
W = W Uj
on P(Fd × Fd , E).
If we define UW = (UW,1 , . . . , UW,d ) on LW by
UW,j : W p → UjR W p = W Sj p,
then, from the intertwining relations
ΦUj = UjR Φ,
ΦUj∗ = Sj
R[∗]
Φ for j = 1, . . . , d
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(2.12)
