Recent advances in matrix and operator theory

339 Pages·2008·1.703 MB·English

by  Joseph A. Ball, Joseph A. Ball, Yuli Eidelman, J. William Helton, Vadim Olshevsky, James Rovnyak

Operator Theory: Advances and Applications
Vol. 179
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. Olshevsky (Storrs)
M. Putinar (Santa Barbara)
Editorial Board:
D. Alpay (Beer-Sheva) L. Rodman (Williamsburg)
J. Arazy (Haifa) J. Rovnyak (Charlottesville)
A. Atzmon (Tel Aviv) D. E. Sarason (Berkeley)
J. A. Ball (Blacksburg) I. M. Spitkovsky (Williamsburg)
A. Ben-Artzi (Tel Aviv) S. Treil (Providence)
H. Bercovici (Bloomington) H. Upmeier (Marburg)
A. Böttcher (Chemnitz) S. M. Verduyn Lunel (Leiden)
K. Clancey (Athens, USA) D. Voiculescu (Berkeley)
L. A. Coburn (Buffalo) D. Xia (Nashville)
R. E. Curto (Iowa City) D. Yafaev (Rennes)
K. R. Davidson (Waterloo, Ontario)
R. G. Douglas (College Station) Honorary and Advisory
A. Dijksma (Groningen) Editorial Board:
H. Dym (Rehovot) C. Foias (Bloomington)
P. A. Fuhrmann (Beer Sheva) T. Kailath (Stanford)
B. Gramsch (Mainz) H. Langer (Vienna)
J. A. Helton (La Jolla) P. D. Lax (New York)
M. A. Kaashoek (Amsterdam)
H. Widom (Santa Cruz)
Subseries: Advances in
Partial Differential Equations
Subseries editors:
Bert-Wolfgang Schulze
Universität Potsdam
Germany
Sergio Albeverio
Universität Bonn
Germany
Recent Advances
in Matrix and
Operator Theory
Joseph A. Ball
Yuli Eidelman
J. William Helton
Vadim Olshevsky
James Rovnyak
Editors
Birkhäuser
. .
Basel Boston Berlin
Editors:
Joseph A. Ball Vadim Olshevsky
Department of Mathematics Department of Mathematics
Virginia Tech University of Connecticut
Blacksburg, VA 24061, USA 196 Auditorium Road, U-9
e-mail: [email protected] Storrs, CT 06269, USA
e-mail: [email protected]
Yuli Eidelman
School of Mathematical Sciences James Rovnyak
Raymond and Beverly Sackler Department of Mathematics
Faculty of Exact Sciences University of Virginia
Tel Aviv University P. O. Box 400137
Ramat Aviv 69978, Israel Charlottesville, VA 22904-4137, USA
e-mail: [email protected] e-mail: [email protected]
J. William Helton
Department of Mathematics
University of California San Diego
9500 Gilman Drive
La Jolla, California 92093-0112, USA
e-mail: [email protected]
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Contents
Editorial Introduction........................................................vii
Daniel Alpay and Israel Gohberg
Inverse Problems for First-Order Discrete Systems........................1
Miha´ly Bakonyi and Kazumi N. Stovall
Stability of Dynamical Systems via Semidefinite Programming...........25
Tom Bella, Vadim Olshevsky and Lev Sakhnovich
Ranks of Hadamard Matrices and Equivalence of Sylvester–Hadamard
and Pseudo-Noise Matrices..............................................35
Yurij M. Berezansky and Artem D. Pulemyotov
Image of a Jacobi Field..................................................47
Vladimir Bolotnikov and Alexander Kheifets
The Higher Order Carath´eodory–JuliaTheorem and Related
Boundary Interpolation Problems.......................................63
Ram´on Bruzual and Marisela Dom´ınguez
A Generalization to Ordered Groups of a Kre˘ınTheorem...............103
Shiv Chandrasekaran, Ming Gu, Jianlin Xia and Jiang Zhu
A Fast QR Algorithm for Companion Matrices.........................111
Nurhan C¸olako˘glu
The Numerical Range of a Class of Self-adjoint Operator Functions.....145
Dario Fasino
A Perturbative Analysis of the Reduction into
Diagonal-plus-semiseparableForm of Symmetric Matrices..............157
Stephan Ramon Garcia
The Eigenstructure of Complex Symmetric Operators..................169
Alexei Yu. Karlovich
Higher Order Asymptotic Formulas for Traces of Toeplitz Matrices
with Symbols in H¨older–Zygmund Spaces..............................185
Sawinder P. Kaur and Israel Koltracht
On an Eigenvalue Problem for Some Nonlinear Transformations
of Multi-dimensional Arrays............................................197
vi Contents
Igor V. Nikolaev
On Embedding of the Bratteli Diagram into a Surface..................211
Vadim Olshevsky, Ivan Oseledets, and Eugene Tyrtyshnikov
Superfast Inversionof Two-Level Toeplitz Matrices Using Newton
Iteration and Tensor-Displacement Structure...........................229
Leiba Rodman and Ilya M. Spitkovsky
On Generalized Numerical Ranges of Quadratic Operators..............241
James Rovnyak and Lev A. Sakhnovich
Inverse Problems for Canonical Differential Equations with
Singularities...........................................................257
Lev A. Sakhnovich
On Triangular Factorization of Positive Operators......................289
Tavan T. Trent
Solutions for the H∞(Dn) Corona Problem Belonging to exp(L2n1−1)...309
Hugo J. Woerdemann
A Matrix and its Inverse: Revisiting Minimal Rank Completions........329
Editorial Introduction
This volume contains the proceedings of the InternationalWorkshopon Operator
TheoryandApplications (IWOTA)whichwasheldatthe UniversityofConnecti-
cut, Storrs, USA, July 24–27, 2005. This was the sixteenth IWOTA; in fact, the
workshop was held biannually since 1981, and annually in recent years (starting
in 2002) rotating among ten countries on three continents. Here is the list of the
fifteen workshops:
IWOTA’1981: Santa Monica, California, USA (J.W. Helton, Chair)
IWOTA’1983: Rehovot, Israel (H. Dym, Chair)
IWOTA’1985: Amsterdam, The Netherlands (M.A. Kaashoek, Chair)
IWOTA’1987: Mesa, Arizona, USA (L. Rodman, Chair)
IWOTA’1989: Rotterdam, The Netherlands (H. Bart, Chair)
IWOTA’1991: Sapporo, Hokkaido, Japan (T. Ando, Chair)
IWOTA’1993: Vienna, Austria (H. Langer, Chair)
IWOTA’1995: Regensburg, Germany (R. Mennicken, Chair)
IWOTA’1996: Bloomington, Indiana, USA (H. Bercovici, C. Foias, Co-chairs)
IWOTA’1998: Groningen, The Netherlands (A. Dijksma, Chair)
IWOTA’2000: Faro, Portugal(A.F. dos Santos, Chair)
IWOTA’2002: Blacksburg, Virginia, USA (J. Ball, Chair)
IWOTA’2003: Cagliari, Italy (S. Seatzu, C. van der Mee, Co-Chairs)
IWOTA’2004: Newcastle upon Tyne, UK (M.A. Dritschel, Chair)
IWOTA’2005: Storrs, Connecticut, USA (V. Olshevsky, Chair)
The aim of the 2005 IWOTA was to review recent advances in operator theory
and its applications to several areas including mathematical systems theory and
control theory.
Amongthemaintopicsoftheworkshopwasthestudyofstructuredmatrices,
theirapplications,andtheirroleinthedesignoffastandnumericallyreliablealgo-
rithms. This topic had already received a considerable attention at IWOTA’2002
and IWOTA’2003 when the main focus was mostly on the structures of Toeplitz,
HankelandPicktypes.Intheyear2005theinterestshiftedtowardsmatriceswith
quasiseparable structure.
The IWOTA’2005wasmade possible throughthe generousfinancialsupport
ofNationalScienceFoundation(award:0536873)aswellasthankstothefundsof
theCollegeofArtsandSciencesandofthe ResearchFoundationofthe University
of Connecticut. All this support is acknowledged with a gratitude.
Joseph Ball, Yuli Eidelman, William Helton,
Vadim Olshevsky, and James Rovnyak (Editors)
OperatorTheory:
Advances andApplications,Vol.179,1–24
(cid:1)c 2007Birkh¨auserVerlagBasel/Switzerland
Inverse Problems for First-Order Discrete
Systems
Daniel Alpay and Israel Gohberg
Abstract. Westudyinverseproblemsassociatedtofirst-orderdiscretesystems
intherationalcase.Weshowinparticularthateveryrationalfunctionstrictly
positiveontheunitcircleisthespectralfunction ofsuchasystem.Formulas
for the coefficients of the system are given in terms of realizations of the
spectral function or in termsof arealization ofa spectral factor. Theinverse
problemsassociatedtothescatteringfunctionandtothereflectioncoefficient
function are also studied. An important role in the arguments is played by
the state space method. We obtain formulas which are very similar to the
formulas we have obtained earlier in the continuous case in our study of
inverseproblems associated to canonical differential expressions.
Mathematics Subject Classification (2000). Primary: 34A55, 49N45, 70G30;
Secondary: 93B15, 47B35.
Keywords.Inverseproblems,spectralfunction,scatteringfunction,Schurpa-
rameters, state space method.
1. Introduction
Here we continue to study first-order discrete systems. We defined the character-
istic spectral functions associated to a first-order discrete in [7] and studied the
correspondinginverseproblemsin[8]forscalarsystems.Inthematrix-valuedcase,
see [3], a system of equations of the form
(cid:1) (cid:2) (cid:1) (cid:2)
∗
I α zI 0
Xn(z)= βp In 0p I Xn−1(z), n=1,2,..., (1.1)
n p p
iscalledacanonicaldiscretefirst-orderone-sidedsystem.Thesequenceofmatrices
(α ,β )isnotarbitrary,buthasthe followingproperty:thereexistsasequence∆
n n
Daniel Alpaywishes tothank the EarlKatzfamilyforendowing the chairwhichsupported his
research.
2 D. Alpay and I. Gohberg
of strictly positive block diagonal matrices in C2p×2p such that
(cid:1) (cid:2) (cid:1) (cid:2)
∗
I α I α
βp In J∆n βp In =J∆n−1, n=1,2,..., (1.2)
n p n p
where (cid:1) (cid:2)
I 0
J = p .
0 −I
p
The sequence is then called ∆-admissible. In the scalar case (that is, when p=1)
condition (1.2) forces α = β∗ (see [3]). Still for p = 1 these systems arise as the
n n
discretization of the telegrapher equation; see [7] for a discussion and references.
Ana posteriori motivationforthestudyofsuchsystemsisthe factthatweobtain
formulas very close to the ones we proved in the continuous case in our study
of inverse problems associated to canonical differential expressions. To be more
precise we need to present our setting in greater details. We first gather the main
results from [3] needed in the sequel. Let
(cid:1) (cid:2) (cid:1) (cid:2)
zI 0 0 β∗
Z = p and F = n , n=1,2,....
0 I n α∗ 0
p n
Under the hypothesis
(cid:3)∞
((cid:1)α (cid:1)+(cid:1)β (cid:1))<∞, (1.3)
n n
n=1
the infinite product
⎛ ⎞
(cid:1)
(cid:6)∞
Y(z)=⎝ (I +Z−nF Zn)⎠ (1.4)
2p n
n=1
converges absolutely and uniformly on the unit circle, and the functions
(cid:9) (cid:10)
X (z)=Zn (I +Z−nF Zn)···(I +Z−1F Z) Y(z)−1, n=1,2,...,
n 2p n 2p 1
define the unique C2p×2p-valued solution to the system (1.1) with the property
that (cid:1) (cid:2)
z−nI 0
lim p X (z)=I , |z|=1. (1.5)
n→∞ 0 Ip n 2p
See [3, Section 2.1]. This solution is called the fundamental solution of the first-
order discrete system (1.1). The function Y(z)−1 is called the asymptotic equiva-
lence matrix function; see [3, Section 2.2]. Under the supplementary hypothesis
lim ∆ >0 (1.6)
n
n−→∞
the function Y(z) allows to define the characteristic spectral functions of the sys-
tem (1.1). We note that when (1.6) is not in force the situation seems to be much
more involved, and leads to degenerate cases. Furthermore, conditions such as
(1.2) and (1.6) seem to be specific of the discrete case; no counterpart of these
conditions is needed in the continuous case.
Inverse Problems for Discrete Systems 3
Let (cid:1) (cid:2)
δ 0
lim ∆ = 1 . (1.7)
n→∞ n 0 δ2
The function
W(z)=((Y +Y )(1/z))−1δ−1((Y +Y )(1/z))−∗
21 22 2 21 22
=((Y +Y )(1/z))−1δ−1((Y +Y )(1/z))−∗
11 12 1 11 12
is called the spectral function. The Weyl function is the uniquely defined function
N(z) analytic in the closed unit disk such that N(0)=iI and
p
W(z)=Im N(z), |z|=1.
AssociatedtoN isthereproducingkernelspaceoffunctionswithreproducingker-
nel N(z)−N(w)∗ and denoted by L(N). The function W(z) is the spectralfunction
z−w∗
of the unitary operator U defined in L(N) by
f(z)−f(α)
(U −αI)−1f(z)= , |α|(cid:3)=1.
z−α
See [11].
From (1.5) follows that there exists a C2p×p-valued solution B (z) to (1.1) with
n
the following properties:
(cid:9) (cid:10)
(a) I −I B (z)=0, and
(cid:9) p (cid:10)p 0
(b) 0 I B (z)=I +o(n), |z|=1.
p n p
It then holds that (cid:9) (cid:10)
I 0 B (z)=znS(z)+o(n)
p n
where
S(z)=(Y (z)+Y (z))(Y (z)+Y (z))−1. (1.8)
11 12 21 22
Thefunction(1.8)iscalledthescatteringmatrixfunction associatedtothediscrete
system. The scattering matrix function has the following properties: it is in the
Wiener algebra Wp×p (see the end of the section for the definition), admits a
Wiener–Hopf factorization and is such that
S(z)∗δ S(z)=δ , |z|=1. (1.9)
1 2
See [3, Section 2.3]. The inverse scattering problem considered in this paper is
definedasfollows:givenafunctionS(z)whichadmitsaWiener–Hopffactorization
and satisfies moreoverthe condition(1.9) for some matrices δ and δ , is S(z)the
1 2
scattering function of a first-order discrete system?
Some preliminary notation and remarks are needed to define the reflection coeffi-
cient function. First, for
(cid:1) (cid:2)
M M
M = 11 12 ∈C2p×2p and X ∈Cp×p
M M
21 22
we define the linear fractional transformation T (X):
M
T (X)=(M X+M )(M X +M )−1.
M 11 12 21 22