Recent Advances in Operator Theory and Its Applications: The Israel Gohberg Anniversary Volume

479 Pages·2005·4.19 MB·English

by  Marinus A. Kaashoek

Operator Theory: Advances and
Applications
Vol. 160
Editor:
I. Gohberg
Editorial Office: H. G. Kaper (Argonne)
School of Mathematical S. T. Kuroda (Tokyo)
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)
D. Alpay (Beer-Sheva) J. Rovnyak (Charlottesville)
J. Arazy (Haifa) D. E. Sarason (Berkeley)
A. Atzmon (Tel Aviv) I. M. Spitkovsky (Williamsburg)
J. A. Ball (Blacksburg) S. Treil (Providence)
A. Ben-Artzi (Tel Aviv) H. Upmeier (Marburg)
H. Bercovici (Bloomington) S. M. Verduyn Lunel (Leiden)
A. Böttcher (Chemnitz) D. Voiculescu (Berkeley)
K. Clancey (Athens, USA) H. Widom (Santa Cruz)
L. A. Coburn (Buffalo) D. Xia (Nashville)
K. R. Davidson (Waterloo, Ontario) D. Yafaev (Rennes)
R. G. Douglas (College Station)
A. Dijksma (Groningen) Honorary and Advisory
H. Dym (Rehovot) Editorial Board:
P. A. Fuhrmann (Beer Sheva) C. Foias (Bloomington)
B. Gramsch (Mainz) P. R. Halmos (Santa Clara)
G. Heinig (Chemnitz) T. Kailath (Stanford)
J. A. Helton (La Jolla) P. D. Lax (New York)
M. A. Kaashoek (Amsterdam) M. S. Livsic (Beer Sheva)
Recent Advances in
Operator Theory
and its Applications
The Israel Gohberg Anniversary Volume
International Workshop on Operator Theory and its Applications
IWOTA 2003, Cagliari, Italy
Marinus A. Kaashoek
Sebastiano Seatzu
Cornelis van der Mee
Editors
Birkhäuser Verlag
. .
Basel Boston Berlin
Editors:
Marinus A. Kaashoek Sebastiano Seatzu
Department of Mathematics, FEW Cornelis van der Mee
Vrije Universiteit Dipartimento di Matematica
De Boelelaan 1081A Università di Cagliari
1081 HV Amsterdam Viale Merello 92
The Netherlands 09123 Cagliari
e-mail: [email protected] Italy
e-mail: [email protected]
[email protected]
2000 Mathematics Subject Classification 34, 35, 45, 47, 65, 93
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Library of Congress, Washington D.C., USA
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ISBN 3-7643-7290-7 Birkhäuser Verlag, Basel – Boston – Berlin
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Contents
Editorial Preface .......................................................... vii
T. Aktosun, M.H. Borkowski, A.J. Cramer and L.C. Pittman
Inverse Scattering with Rational Scattering Coefficients and
Wave Propagationin Nonhomogeneous Media ....................... 1
T. Ando
Aluthge Transforms and the Convex Hull of the Spectrum
of a Hilbert Space Operator ......................................... 21
W. Bhosri, A.E. Frazho and B. Yagci
Maximal Nevanlinna-Pick Interpolation for Points
in the Open Unit Disc ............................................... 41
M.R. Capobianco, G. Criscuolo and P. Junghanns
On the Numerical Solution of a Nonlinear Integral Equation
of Prandtl’s Type .................................................... 53
M. Cappiello
Fourier Integral Operators and Gelfand-Shilov Spaces ................ 81
D.Z. Arov and H. Dym
Strongly Regular J-inner Matrix-valued Functions
and Inverse Problems for Canonical Systems ......................... 101
C. Estatico
Regularization Processes for Real Functions
and Ill-posed Toeplitz Problems ..................................... 161
K. Galkowski
Minimal State-space Realization for a Class of nD Systems .......... 179
G. Garello and A. Morando
Continuity in Weighted Besov Spaces for Pseudodifferential
Operators with Non-regular Symbols ................................ 195
G.J. Groenewald and M.A. Kaashoek
A New Proof of an Ellis-Gohberg Theorem on Orthogonal
Matrix Functions Related to the Nehari Problem .................... 217
vi Contents
G. Heinig and K. Rost
Schur-type Algorithms for the Solution of Hermitian
Toeplitz Systems via Factorization ................................... 233
M. Kaltenba¨ck, H. Winkler and H. Woracek
Almost PontryaginSpaces ........................................... 253
D.S. Kalyuzhny˘ı-Verbovetzki˘ı
Multivariable ρ-contractions ......................................... 273
V. Kostrykin and K.A. Makarov
The Singularly Continuous Spectrum and Non-Closed
Invariant Subspaces ................................................. 299
G. Mastroianni, M.G. Russo and W. Themistoclakis
Numerical Methods for Cauchy Singular Integral Equations
in Spaces of Weighted Continuous Functions ......................... 311
A. Oliaro
On a Gevrey-Nonsolvable Partial Differential Operator ............... 337
V. Olshevsky and L. Sakhnovich
Optimal Prediction of Generalized Stationary Processes .............. 357
P. Rocha, P. Vettori and J.C. Willems
Symmetries of 2D Discrete-Time Linear Systems ..................... 367
G. Rodriguez, S. Seatzu and D. Theis
An Algorithm for Solving Toeplitz Systems by Embedding
in Infinite Systems ................................................... 383
B. Silbermann
Fredholm Theory and Numerical Linear Algebra ..................... 403
C.V.M. van der Mee and A.C.M. Ran
Additive and Multiplicative Perturbations of Exponentially
Dichotomous Operators on General Banach Spaces ................... 413
C.V.M. van der Mee, L. Rodman and I.M. Spitkovsky
Factorization of Block Triangular Matrix Functions
with Off-diagonal Binomials ......................................... 425
G. Wanjala
Closely Connected Unitary Realizations of the Solutions to the
Basic Interpolation Problem for Generalized Schur Functions ......... 441
M.W. Wong
Trace-Class Weyl Transforms ........................................ 469
Editorial Preface
Thisvolumecontainsaselectionofpapersinmodernoperatortheoryanditsappli-
cations. Most of them are directly related to lectures presented at the Fourteenth
InternationalWorkshop on Operator Theory and its Applications (IWOTA 2003)
held at the University of Cagliari,Italy, in the period of June 24–27, 2003.
The workshop, which was attended by 108 mathematicians – including a
number of PhD and postdoctoral students – from 22 countries, presented eight
special sessions on
1) control theory,
2) interpolation theory,
3) inverse scattering,
4) numerical estimates for operators,
5) numerical treatment of integral equations,
6) pseudodifferential operators,
7) realizations and transformations of analytic functions and indefinite inner
product spaces, and
8) structured matrices.
The program consisted of 19 plenary lectures of 45 minutes and 78 lectures of 30
minutes in four parallel sessions.
The present volume reflects the wide range and rich variety of topics pre-
sentedanddiscussedattheworkshop,bothwithinandoutsidethespecialsessions.
The papers deal with inverse scattering, numerical ranges, pseudodifferential op-
erators,numericalanalysis,interpolation theory, multidimensional system theory,
indefinite inner products, spectral factorization, and stationary processes.
SinceintheperiodthattheproceedingsofIWOTA2003werebeingprepared,
Israel Gohberg, the president of the IWOTA steering committee, reached the age
of 75, we decided to dedicate these proceedings to Israel Gohberg on the occasion
of his 75th birthday.Alloftheauthorsoftheseproceedingshavejoinedtheeditors
and dedicated their papers to him as well.
The Editors
Israel Gohberg, the president
of the IWOTA steering committee
OperatorTheory:
Advances andApplications,Vol.160,1–20
(cid:1)c 2005Birkh¨auserVerlagBasel/Switzerland
Inverse Scattering with Rational Scattering
Coefficients and Wave Propagation in
Nonhomogeneous Media
Tuncay Aktosun, Michael H. Borkowski, Alyssa J. Cramer and Lance
C. Pittman
Dedicated toIsrael Gohberg on the occasion of his 75th birthday
Abstract. Theinversescatteringproblemfortheone-dimensionalSchr¨odinger
equationisconsideredwhenthepotentialisrealvaluedandintegrableandhas
afinitefirst-moment andno boundstates. Corresponding to suchpotentials,
forrationalreflectioncoefficientswithonlysimplepolesintheupperhalfcom-
plex plane, amethod is presented to recoverthepotential and thescattering
solutions explicitly.A numericalimplementation of themethod is developed.
For such rational reflection coefficients, the scattering wave solutions to the
plasma-wave equation are constructed explicitly. The discontinuities in these
wavesolutionsandintheirspatialderivativesareexpressedexplicitlyinterms
of the potential.
Mathematics SubjectClassification (2000). Primary 34A55; Secondary 34L40
35L05 47E05 81U40.
Keywords.Inversescattering,Schro¨dingerequation,Rationalscatteringcoef-
ficients, Wavepropagation, Plasma-wave equation.
1. Introduction
Consider the Schro¨dinger equation
ψ(cid:1)(cid:1)(k,x)+k2ψ(k,x)=V(x)ψ(k,x), x∈R, (1.1)
The research leading to this paper was supported by the National Science Foundation under
grants DMS-0243673 and DMS-0204436 and by the Department of Energy under grant DE-
FG02-01ER45951.
2 T. Aktosun, M.H. Borkowski,A.J. Cramer and L.C. Pittman
where the prime denotes the x-derivative, and the potential V is assumed to have
no bound states and to belong to the Faddeev class. The bound states of (1.1)
correspondtoitssquare-integrablesolutions.Bythe(cid:1)Faddeevclasswemeantheset
ofreal-valuedandmeasurablepotentialsfor which ∞ dx(1+|x|)|V(x)| is finite.
−∞
Via the Fourier transformation
(cid:2)
∞
1
u(x,t)= dkψ(k,x)e−ikt,
2π
−∞
we can transform (1.1) into the plasma-waveequation
∂2u(x,t) ∂2u(x,t)
− =V(x)u(x,t), x,t∈R. (1.2)
∂x2 ∂t2
In the absence of bound states, (1.1) does not have any bounded solutions
for k2 < 0. The solutions for k2 > 0 are known as the scattering solutions. Each
scattering solution can be expressed as a linear combination of the two (linearly-
independent) Jost solutions from the left and the right, denoted by fl and fr,
respectively, satisfying the respective asymptotic conditions
fl(k,x)=eikx[1+o(1)], fl(cid:1)(k,x)=ikeikx[1+o(1)], x→+∞,
fr(k,x)=e−ikx[1+o(1)], fr(cid:1)(k,x)=−ike−ikx[1+o(1)], x→−∞.
We have
1 L(k)
fl(k,x)= eikx+ e−ikx+o(1), x→−∞,
T(k) T(k)
1 R(k)
fr(k,x)= e−ikx+ eikx+o(1), x→+∞,
T(k) T(k)
where L and R are the left and right reflection coefficients, respectively, and T is
the transmission coefficient.
Thesolutionsto(1.1)fork =0requirespecialattention.Generically,fl(0,x)
and fr(0,x) are linearly independent on R, and we have
T(0)=0, R(0)=L(0)=−1.
In the exceptional case, fl(0,x) and fr(0,x) are linearly dependent on R and we
have (cid:3)
T(0)= 1−R(0)2 >0, −1<R(0)=−L(0)<1.
When V belongs to the Faddeev class and has no bound states, it is known
[1–5]thateitheroneofthereflectioncoefficientsRandLcontainstheappropriate
informationtoconstructtheotherreflectioncoefficient,thetransmissioncoefficient
T, the potential V, and the Jost solutions fl and fr. Our aim in this paper is to
present explicit formulas for such a construction when the reflection coefficients
arerationalfunctions ofk with simple poles onthe upper half complex planeC+.
We will use C− to denote the lower half complex plane and let C+ := C+ ∪R
and C− :=C−∪R.
The recovery of V from a reflection coefficient constitutes the inverse scat-
teringproblemfor(1.1).Therehasbeenasubstantialamountofpreviouswork[2,