The Gohberg Anniversary Collection: Volume II: Topics in Analysis and Operator Theory

540 Pages·1989·15.09 MB·English

by  Frederic Abergel

OT41
Operator Theory: Advances and Applications
Vol. 41
Editor:
I. Gobberg
Tel Aviv University
RamatAviv, Israel
Editorial Office:
School of Mathematical Sciences
Tel Aviv University
RamatAviv, Israel
Editorial Board:
A. Atzmon (Tel Aviv) M. A. Kaashoek (Amsterdam)
J. A. Ball (Blacksburg) T. Kailath (Stanford)
L. de Branges (West Lafayette) H. G. Kaper (Argonne)
K. Clancey (Athens, USA) S. T. Kuroda (Tokyo)
L. A. Coburn (Buffalo) P. Lancaster (Calgary)
R. G. Douglas (Stony Brook) L. E. Lerer (Haifa)
H. Dym (Rehovot) E. Meister (Darmstadt)
A. Dynin (Columbus) B. Mityagin (Columbus)
P. A. Fillmore (Halifax) J. D. Pincus (Stony Brook)
C. Foias (Bloomington) M. Rosenblum (Charlottesville)
P. A. Fuhrmann (Beer Sheva) J. Rovnyak (Charlottesville)
S. Goldberg (College Park) D. E. Sarason (Berkeley)
B. Gramsch (Mainz) H. Widom (Santa Cruz)
J. A. Helton (La Jolla) D. Xia (Nashville)
D. Herrero (Tempe)
Honorary and Advisory
Editorial Board:
P. R. Halmos (Santa Clara) S. G. Mikhlin (Leningrad)
T. Kato (Berkeley) R. Phillips (Stanford)
P. D. Lax (New York) B. Sz.-Nagy (Szeged)
M. S. Livsic (Beer Sheva)
Birkhauser Verlag
Basel· Boston· Berlin
The Gohberg Anniversary
Collection
Volume II: Topics in Analysis and
Operator Theory
Edited by
H.Dym
S. Goldberg
M. A. Kaashoek
P. Lancaster
1989 Birkhauser Verlag
Basel· Boston· Berlin
Volume Editorial Office:
Department of Mathematics
and Computer Science
Vrije Universiteit
Amsterdam, The Netherlands
CIP-Titelaufnahme der Deutschen Bib60thek
The Gohberg anniversary collection / [vol. ed. office: Dep. of
Mathematics and Computer science, Vrije Univ., Amsterdam,
The Netherlands]. Ed. by H. Dym ... - Basel; Boston; Berlin
Birkhiiuser
(Operator theory; ... )
ISBN 3-7643-2283-7 (Basel ... )
ISBN 0-8176-2283-7 (Boston)
NE: Dym, Harry [Hrsg.]; Vrije Universiteit <Amsterdam> / FacuIteit
derWiskunde en Informatica; Gochberg, Izrail': Festschrift
Vol. 2. Topics in analysis and operator theory. - 1989
Topics in analysis and operator theory / [vol. ed. office: Dep.
of Mathematics and Computer Science, Vrije Univ.,
Amsterdam, The Netherlands]. Ed. by H. Dym ... - Basel
Boston ; Berlin : Birkhiiuser, 1989
(The Gohberg anniversary collection ; Vol. 2)
(Operator theory ; Vol. 41)
ISBN 3-7643-2308-6 (Basel ... ) Pb.
ISBN 0-8176-2308-6 (Boston) Pb.
NE: Dym, Harry [Hrsg.]; 2. GT
This work is subject to copyright. All rights are reserved, whether the whole or part
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made for other than private use a fee is payable to >NerwertungsgesellschaftWort«, Munich.
© 1989 Birkhiiuser Verlag Basel
Softcover reprint of the hardcover 1s t edition 1989
ISBN-I3: 978-3-0348-9975-8 e-ISBN-I3: 978-3-0348-9278-0
001: 10.1007/978-3-0348-9278-0
v
Table of contents of Volume II
Portrait I. Gohberg ........................................ VII
Editorial Preface . . . . . . . IX
Abergel F., Temam R.:
Duality methods for the boundary control of some evolution equations 1
ArocenaR.:
Unitary extensions of isometries and contractive intertwining dilations 13
BallI.A., Helton I. W:
Factorization and general properties of nonlinearToeplitz operators 25
Baumgiirtel H. :
Quasilocal algebras over index sets with a minimal condition . . . . . 43
Bercovici H., Voiculescu D.:
The analogue of Kuroda's theorem for n-tuples 57
Clancey K. F.:
The geometry of representing measures and their critical values ... 61
Costabel M., Saranen I. :
Boundary element analysis of a direct method for the biharmonic Dirichlet
problem .............................................. 77
Cotlar M., Sadosky C. :
Nonlinear lifting theorems, integral representations and stationary processes in
algebraic scattering systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Curgus B., DijksmaA., Langer H., Snoo H.S. V de:
Characteristic functions of unitary colligations and of bounded operators in
Krein spaces ........................................... 125
Djrbashian M. M.:
Differential operators of fractional order and boundary value problems in the
complex domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
DymH.:
On reproducing kernel spaces, J unitary matrix functions, interpolation and
displacement rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
FeintuchA.:
On asymptoticToeplitz and Hankel operators 241
c.,
Foias Tannenbaum A. :
Iterative commutant lifting for systems with rational symbol 255
Frank L. S., Heijstek I.J.:
On the reduction of coercive singular perturbations to regular perturbations . . . 279
VI
Greenberg W, Polewczak I.:
Averaging techniques for the transport operator and an existence theorem for
the BGK equation in kinetic theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Helton I. W:
Factorization of nonlinear system 311
KaashoekM.A., WoerdemanH.I.:
Minimal lower separable representations: characterization and construction . 329
Kerchy L.:
On the inclination of hyperinvariant subspaces of C 11-contractions 345
Korenblum B.:
Unimodular Mobius-invariant contractive divisors for the Bergman space 353
McLeanW, WendlandWL.:
Trigonometric approximation of solutions of periodic pseudodifferential
equations ............................................. 359
Meister E., Speck F.-D.:
Wiener-Hopf factorization of certain non-rational matrix functions in mathe-
matical physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
NakamuraY:
Classes of operator monotone functions ans Stieitjes functions 395
Nikolskii N.K., Vasyunin VI.:
A unified approach to function models, and the transcription problem. . . . . . . 405
Prossdorf S., Rathsfeld A.:
Quadrature methods for strongly elliptic Cauchy singular integral equations on
an interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
Santos A. F. dos:
General Wiener-Hopf operators and representation of their generalized inver-
ses ............................. . 473
SarasonD.:
Exposed points in HI, I .................................... 485
Semenov E.M., Shneiberg I. Th.:
Geometrical properties of a unit sphere of the operator spaces in Lp . . . . . . . . 497
Taylor K. F. :
C* -algebras of Crystal groups ................................ 511
WidomH.:
On Wiener-Hopf determinants 519
Table of contents of Volume I ...................... . . . . . . . . . . . . 545
Errata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
VII
Israel Gohberg
IX
Editorial Preface
The Gohberg Anniversary Collection is dedicated to Israel Gohberg.lt contains the
proceedings of the international conference on "Operator Theory: Advances and Applications",
which was held in Calgary, August 22 -26,1988, on the occasion of his 60th birthday. The two
volumes present an uptodate and attractive account of recent advances in operator theory, matrix
theory and their applications. They include contributions by a number of the world's leading
specialists in these areas and cover a broad spectrum of topics, many of which are on the frontiers
of reseach. The two volumes are a tribute to Israel Gohberg; they reflect the wide range of his
mathematical initiatives and personal interests.
This second volume consists of original research papers on linear operator theory, on
nonlinear Toeplitz operators and nonlinear lifting theorems, on complex function theory, on
numerical analysis, and on applications ofo perator theory to problems in analysis, in the theory of
differential equations, in control theory, and in mathematical physics. Recent theoretical
developments are presented as well as new advances in the study of specific classes of operators.
The Editors are grateful to the Department ofM athematics and Computer Science of the
Vrije Unversiteit for providing secretarial assistance. In particular, they thank Marijke Titawano for
her work.
The Editors
Operator Theory:
Advances and Applications, Vol. 41
© 1989 Birkhiiuser Verlag Basel
DUALITY METHODS FOR THE BOUNDARY CONTROL OF SOME EVOLUTION
EQUATIONS.
Frederic Abergel and Roger Temam
Dedicated to Israel Gohberg on the occasion of his 60th birthday
We present some new and simple proofs, using convex analysis, for
some results related to the boundary control of parabolic and hyperbolic
evolution equations.
INTRODUCTION
The quadratic cost problem for the boundary control of evolution
equations has been extensively studied, and, thanks to recent regularity
results [L-2) - [L-T) - [L-L-T) , its study has made important progress. The
main results concerning these problems are to be found in [L-l) - [L-2) -
[L-T) , in which the optimality conditions are derived, some regulartiy
results for the optimal control are proven, and the existence of a pointwise
feedback operator is established.
It has appeared to us that, for the first part of the results
evoked above, i.e. the existence and regularity of the optimal control as
well as the system of optimality conditions, the use of classical duality
methods in Convex Analysis provides us with straightforward and simpler
proofs of these results; we therefore find it interesting to expose how these
mehtods apply for such problems.
We shall deal with the boundary control problem for parabolic and
hyperbolic dynamics. In the parabolic case, we recover the existence and
regularity results of [L-2) , and our proofs are essentially self contained.
In the hyperbolic case, we have to restrain ourselves to the finite time
2 Abergel and Temam
interval problem, recovering only partly the results of [L-T]; moreover, our
proofs rely crucially on the recent regularity results in [L-L-T].
In the first two sections, we are interested in the parabolic
dynamics:
(it -
~) y = 0 in ~ = 0 x (O,T),
{
(1.1) . y =uonLr=rx(O,T),
yeo) = Yoin 0,
where 0 is a bounded open set of Rn,N ~ 2, with a smooth boundary r, y
o
is given in L2(O) , and the boundary control u is in L2(~). T is an
(extended) real number, strictly larger than zero. We are interested in the
quadratic cost problem:
0 find U in L2 (Et),minimizing the cost function
[l:eu) = ~ lIy(u)1I22 + ~ lIull22
L (~) L (~)
with y(u) solution of (1.1)
Our results will be obtained by using duality methods [E-T]
for (PT) , and are summed up in the
THEOREM A Let u be in L2(~), y(u) be the solution of (1.1);
(a) (u,y(u» is an optimal pair if and only if there exists f in L2(~)
such that
y(u) = -f in ~ (A.I)
a
u = 7Jii (v(f» on ~ (A.2),
where v(f) is the solution of the adjoint evolution problem: