Reproducing Kernel Spaces and Applications

354 Pages·2003·7.585 MB·English

by  J. Agler, F. B. Yeh, N. J. Young (auth.), Daniel Alpay (eds.)

Operator Theory: Advances and
Applications
VoI. 143
Editor:
1. Gohberg
Editorial Oftice:
School of Mathematical Sciences P. Lancaster (Calgary)
Tel Aviv University L. E. Lerer (Haifa)
Ramat Aviv, Israel B. Mityagin (Columbus)
V. V. Peller (Manhattan, Kansas)
Editorial Board: L. Rodman (Williamsburg)
J. Arazy (Haifa) J. Rovnyak (Charlottesville)
A. Atzmon (Tel Aviv) D. E. Sarason (Berkeley)
J. A. BaII (Blacksburg) 1. M. Spitkovsky (Williamsburg)
A. Ben-Artzi (Tel Aviv) S. Treil (Providence)
H. Bercovici (Bloomington) H. Upmeier (Marburg)
A. Bottcher (Chemnitz) S. M. Verduyn Lunei (Leiden)
K. Clancey (Athens, USA) D. Voiculescu (Berkeley)
L. A. Coburn (Buftalo) H. Widom (Santa Cruz)
K. R. Davidson (Waterloo, Ontario) D. Xia (Nashville)
R. G. Douglas (College Station) D. Yafaev (Ren nes)
H. Dym (Rehovot)
P. A. Fuhrmann (Beer Sheva)
S. Goldberg (College park) Honorary and Advisory
B. Gramsch (Mainz) Editorial Board:
G. Heinig (Chemnitz) C. Foias (Bloomington)
J. A. Helton (La Jolla) P. R. Halmos (Santa Clara)
M. A. Kaashoek (Amsterdam) T. Kailath (Stanford)
H. G. Kaper (Argonne) P. D. Lax (New York)
S. T. Kuroda (Tokyo) M. S. Livsic (Beer Sheva)
Reproducing Kernel Spaces
and Applications
Daniel Alpay
Editor
Springer Basel AG
Editor:
Daniel Alpay
Department of Mathematics
Ben Gurion University of the Negev
P.O. Box 653
Beer Sheva 84105
Israel
e-mail: [email protected]
2000 Mathematics Subject Classification 46E22, 47B32, 47N20, 47N50
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
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ISBN 978-3-0348-9430-2 ISBN 978-3-0348-8071-0 (eBook)
DOI 10.1007/978-3-0348-8071-0
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Contents
Editorial Introduction .................................................... Vll
J. AGLER, F.B. YEH AND N.J. YOUNG
Realization of Functions into the Symmetrised Bidisc ................ 1
D. ALPAY, T.YA. AZIZOV, A. DIJKSMA, H. LANGER AND G. WANJALA
A Basic Interpolation Problem for Generalized Schur Functions
and Co isometric Realizations ........................................ 39
J.A. BALL AND V. VINNIKOV
Formal Reproducing Kernel Hilbert Spaces:
The Commutative and Noncommutative Settings 77
M.F. BESSMERTNYI
On Realizations of Rational Matrix Functions
of Several Complex Variables II ...................................... 135
D.-C. CHANG, R. GILBERT AND J. TIE
Bergman Projection and Weighted Holomorphic Functions 147
H. DYM
Linear Fractional Transformations, Riccati Equations
and Bitangential Interpolation, Revisited ............................ 171
F. GESZTESY AND L.A. SAKHNOVICH
A Class of Matrix-valued Schrodinger Operators
with Prescribed Finite-band Spectra ................................. 213
T.L. KRIETE
Laplace Transforms Asymptotics, Bergman Kernels
and Composition Operators ................................ . . . . . . . . .. 255
M. MBOUP
On the Structure of Self-similar Systems:
A Hilbert Space Approach........................................... 273
S. SAITOH
Reproducing Kernels and a Family of
Bounded Linear Operators ........................................... 303
F.H. SZAFRANIEC
Multipliers in the Reproducing Kernel Hilbert Space,
Subnormality and Noncommutative Complex Analysis 313
F.-H. VASILESCU
Existence of Unitary Dilations as a Moment Problem ................ 333
Operator Theory:
Advances and Applications, Vol. 143, vii-xv
© 2003 Birkhiiuser Verlag Basel/Switzerland
Editorial Introduction
Daniel Alpay
This volume contains a selection of papers that relate to the general topic of
reproducing kernel Hilbert spaces, that is, Hilbert spaces of functions for which
point evaluations are bounded. These spaces appear in a wide range of situations
and they possess additional structure (thanks to the reproducing kernel) that other
Hilbert spaces do not have. This gives, as is illustrated in the papers presented here,
a supplementary point of view that allows one to state and solve new problems.
Let H be a Hilbert space of functions defined on a set n for which the point
evaluations f ~ f(w) are bounded for all wEn. By the Riesz representation
theorem there exists a function k(z,w) defined on n x n with the following two
properties:
1. For every wEn the function kw : z ~ k(z, w) belongs to H.
2. For every wEn and f E H
(I, kW)?-l = f(w).
The function k(z, w) is uniquely defined and is called the reproducing kernel of
the space.
An important feature in a reproducing kernel Hilbert space H is that one can
compute the inner product explicitly on the dense set spanned by the functions
kw : z ~ k(z,w) since
(kw, kvhi = k(v, w).
The reproducing kernel is positive in the sense that all the matrices
k(WI,WI)
k(WI,WN) )
( k(W2,WI) k(W~:~N) 20
k(WN' WI) k(WN,WN)
for every choice of the integer N and of the points WI, ... , W N E n. Conversely each
positive function defines a unique reproducing kernel Hilbert space of functions.
Two typical examples of positive kernels are k(t, s) = min (t, s) with t, s E lR.+
1
(the covariance function of the Brownian motion) and k(z, w) = for z, W
1-zw*
in the open unit disk (the reproducing kernel of the Hardy space of the open unit
disk).
Vlll D. Alpay
The notion of positivity can be traced back to the work of Mercer [47] at the
beginning of the X Xth century, the motivation arising from the work of Hilbert
on integral equations; see Stewart's paper [61, p. 422-423] which contains a quite
complete account of the history of positivity. We also send the reader to the notes
and remarks on pages 84-85 of [20]. The notion of reproducing kernel seems to
have appeared first in a paper of Zaremba and is more than 90 years old; see [66].
The papers of N. Aronszajn [14] and of L. Schwartz [59] are classics that can be
studied repeatedly with much gain.
The main result of the theory, namely the one-to-one correspondence between
positive functions and reproducing kernel Hilbert spaces, relates operator theory
and the theory of functions. The notions of reproducing kernel and of positivity
proved to be quite useful in numerous fields, of which we mention:
1. Partial differential equations. See [22].
2. Conformal mapping. See [21], [49].
3. Gaussian processes. Every positive function on a set T is the covariance
function of a centered gaussian process indexed by T. See, e.g., [42, Theorem
8.2 p. 117] and [50, p. 79 and p. 93]. When T is a homogeneous tree one is
lead to very interesting problems of harmonic analysis; see [43].
4. Harmonic analysis on semigroups. See [20].
5. The notion of hypergroup and harmonic analysis on hypergroups. See [25].
6. Operator theory. We refer in particular to the work of L. de Branges [27] and
of L. de Branges and J. Rovnyak [28], [29] for work in the sixties and to [18].
7. Inverse problems. We refer in particular to the series of papers of D. Arov
and H. Dym [15], [16], [17].
8. Orthogonal polynomials. See [30] for a recent account.
9. Zero counting problems. See [39], [36].
10. From the 1950' a new partner came into play: the theory of dissipative linear
systems; see [3] for a review.
11. The Schur algorithm. See [7], [4].
12. Interpolation. See [35].
13. Inequalities. See [12].
14. Integral transforms. See [56], [57].
15. The theory of bounded symmetric domains. See [41], [13].
16. Generalizations of a formula of Kac (that in turn is a version of the Szego
formula for the growth of the determinants of blocks of a Toeplitz matrix) for
the growth of the determinant (or traces) of finite sections of a convolution
operator. See [34].
17. The basis problem. See [51, Chapter 4].
18. Approximation in reproducing kernel Hilbert spaces, see [33], and the theory
of spline functions. See [24], [19].
19. Multiresolution analysis and the reproducing kernel particle method. We refer
to the work of W.K. Liu; see [46], [45].
Editorial Introduction IX
20. Pattern recognition and statistical learning theory (the theory of support
vector machines). See [40], [58]. In this last volume we refer in particular to
the papers [63] and [64]. Since this topic is maybe less known to the operator
theory community we mention that the support vector method is a general
approach to function estimation problems. See [63, p. 26].
We note that the above list and the given references are by no way exhaustive. We
refer to the first section of the paper of S. Saitoh in the present volume for another
(and mainly different) list of topics where reproducing kernel spaces appear.
Quite often a given question is best understood in a reproducing kernel
Hilbert space (for instance when using Cauchy's formula in the Hardy space H2)
and one finds oneself as Mr Jourdain of Moliere' Bourgeois Gentilhomme speaking
Prose without knowing it [48, p. 51]:
Par ma foil il y a plus de quarante ans que je dis de la prose sans que
j'en susse rien.l
The reproducing kernel setting allows one to attack more general situations and
understand better the problem under study. As an example, let us mention the
Beurling spaces N = sH2 where s is an inner function. For such a space H2 8
N = H2 8 sH2 is the reproducing kernel Hilbert space with reproducing ker
nel ks(z, w) = I-~~l~(,;v)·. Note that H2 8 sH is invariant under the backward
2
shift operator Rof(z) = f(z)~f(O). Now the kernel ks(z, w) is positive for a wider
class of functions than inner functions, namely for all fUllctions analytic and con
tractive in the open unit disk. The characterization of reproducing kernel Hilbert
spaces with reproducing kernel of the form ks(z, w) leads to a generalization of the
Beurling-Lax theorem due to de Branges and Rovnyak when isometric inclusions
are replaced by contractive inclusions. See [28], [6]. This approach led to recent
progress in the study of the counterparts of the spaces with reproducing kernel
ks(z, w) in the ball; see [5].
Another example of importance is taken from the theory of stochastic pro
cesses. If x(n) is a second order real stationary process with correlation function
E(x(n)x(m)) = r(n - m) then
kx(z, w) = ~ znw*m E(x(n)x(m)) = cp(~) + cp(w)* ,
~ -zw*
n,m=O
where E denotes the mathematical expectation, z, ware in the open unit disk,
and
L00
cp(z) = r(O) + 2 r(n)zn.
I
IOn my conscience, I have spoken prose above these forty years, without knowing anything of
the matter. See [53, p. 234] for the translation.
x D. Alpay
If no hypothesis is made on the stochastic process (besides being second order)
the function kx(z, w) is still positive but has no special additional structure. The
associated reproducing kernel Hilbert space seems then of little use. Kailath and
Lev-Ari introduced the family of a-stationary processes. For these the function
kx(z, w) is of the form
kx(z, w) = a(z) { ~(~) ~ :~~)*} a(w)*,
where a an ~ are possibly matrix-valued functions. The case a(z) == 1 corresponds
to classical stationary processes. See [44]. In this context, it is also worth mention
ing that the theory of Hilbert spaces of entire functions of de Branges [27] serves
to solve the prediction problem for continuous stationary processes. See [38].
Another and last example is the case of finite-dimensional spaces. Any finite
dimensional space of functions is trivially a reproducing kernel Hilbert space and
one would think that this is the end of the story. Just the opposite is true, it is
the beginning of a fascinating story, which has implications in the theory of fast
algorithms (and connects then with the above example of a-stationary processes)
and matrix theory. See [37].
Two recent new trends which connect with reproducing kernel spaces are
several complex variables and non commutativity; examples of reproducing kernel
Hilbert space of several complex variables exist for a long time (for instance the
Hardy space of the ball, see [55], and kernels appearing in the more general theory
of bounded symmetric domains, see [41], [13]) but we have in mind the work of
Agler, see for example [1], [2], and Popescu, see [52]. We would like to mention
also the theory of reproducing kernel spaces of hyperholomorphic functions. This
is also a non commutative theory, which has been developed for a long time (see
[32], [26]), but for which much remains to be done; see [10], [11]. We refer to [62]
for an introduction to hypercomplex analysis.
Of course not every Hilbert space is a reproducing kernel Hilbert space, and
even in a reproducing kernel Hilbert space, reproducing kernel methods are one
way among others, and alternative routes exist. To quote Albert Camus on a
different subject:
I1 n'est pas possible, d'ailleurs, de fonder une attitude sur une emotion
privilegiee. Le sentiment de l'absurde est un sentiment parmi d'autres.
Qu'il ait donne sa couleur a tant de pensees et d'actions prouve seule
ment sa puissance et sa Jegitimite. Mais l'intensite d'un sentiment n'en
trafne pas qu'il soit universel. 2
See [31, p. 20]. On the other hand, it is worth mentioning that, if one is willing
to work with a weakened version of the axiom of choice (in particular in Solovay's
2Besides, it is not possible to base an attitude on a privileged emotion. The feeling of absurd is
one feeling among others. That it gave its print to so many thoughts and actions proves only its
power and its legitimacy. But the intensity of a feeling does not entail that it is universal.
Editorial Introduction Xl
model, in which all sets of real numbers are Lebesgue measurable and the axiom
of choice for countable collections holds; see [60]), then all everywhere defined op
erators in Hilbert spaces are bounded (see [65]) and all Hilbert spaces offunctions
are then reproducing kernel Hilbert spaces. See [9] for an example of a Hilbert
space of functions which is not a reproducing kernel Hilbert space.
The papers presented in this volume cover some of the various aspects of
reproducing kernel spaces described above. More precisely:
1. One complex variable theory: The papers of Chang-Gilbert-Tie, Dym, Kri
ete and Saitoh deal with problems in operator theory in one complex variable.
So does also the paper by Azizov, Dijksma, Langer, Wanjala and the editor
of this volume. Topics touched upon include composition operators, Bergman
spaces, geometrical aspects of reproducing kernel spaces, bitangential inter
polation, and the Schur algorithm.
2. Differential operators: Reproducing kernel spaces and differential operators
are closely related, in particular via the Weyl-Titchmarsh function. The pa
per of F. Gesztesy and L. Sakhnovich explores this direction. The paper of
Mboup deals with the theory of self-similar systems and uses the theory of
Hilbert spaces of entire functions of de Branges and the underlying string
and differential operator.
3. Several complex variables: We first present a translation of a part of the thesis
of M.F. Bessmertnyi [23]. A translation of another part of this thesis appears
in the volume [8]. The paper of Agler-Yeh-Young explores one of the recent
new directions in the passage from the one complex variable operator theory
to the several complex variables operator theory, namely realization theory
in the symmetrized bidisc. The paper of Vasilescu studies unitary dilations
for tuples of commuting contractions via moment problems.
4. The noncommutative case: This is considered in the paper of Ball and Vin
nikov (who consider the notion of reproducing kernel in the noncommutative
setting) and in the paper of F. Szafraniec who considers operators of multipli
cation and their subnormality as a background for noncommutative complex
analysis.
Acknowledgments: It is always a pleasure to acknowledge a debt and to thank a
person. A few sentences will not be sufficient to express my gratitude to Harry Dym
and Israel Gohberg for their help and collaboration over the years, but more than
a few words would be more like a confession al a Rousseau [54]. So I will only thank
Israel for inviting me to edit the present volume and Harry for introducing me to
the theory of reproducing kernel spaces. If I recall correctly, the first assignment
he gave me as a PhD student was to read and understand de Branges's book [27].
I am not sure that I have completed the assignment as of now.
Finally it is a pleasure to thank Professor Uri Abraham from Ben-Gurion
University for discussions on Solovay's model.