Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes: A Collection of Papers Dedicated to Lev Sakhnovich

396 Pages·2015·3.354 MB·English

by  Daniel Alpay, Bernd Kirstein (eds.)

Operator Theory
Advances and Applications
244
Daniel Alpay
Bernd Kirstein
Editors
Recent Advances
in Inverse Scattering,
Schur Analysis and
Stochastic Processes
A Collection of Papers
Dedicated to
Lev Sakhnovich

Operator Theory: Advances and Applications
Volume 244

Founded in 1979 by Israel Gohberg

Editors:
Joseph A. Ball (Blacksburg, VA, USA)
Harry Dym (Rehovot, Israel)
Marinus A. Kaashoek (Amsterdam, The Netherlands)
Heinz Langer (Wien, Austria)
Christiane Tretter (Bern, Switzerland)

Associate Editors: Honorary and Advisory Editorial Board:
Vadim Adamyan (Odessa, Ukraine) Lewis A. Coburn (Buffalo, NY, USA)
Wolfgang Arendt (Ulm, Germany) Ciprian Foias (College Station, TX, USA)
Albrecht Böttcher (Chemnitz, Germany) J.William Helton (San Diego, CA, USA)
B. Malcolm Brown (Cardiff, UK) Thomas Kailath (Stanford, CA, USA)
Raul Curto (Iowa, IA, USA) Peter Lancaster (Calgary, Canada)
Fritz Gesztesy (Columbia, MO, USA) Peter D. Lax (New York, NY, USA)
Pavel Kurasov (Stockholm, Sweden) Donald Sarason (Berkeley, CA, USA)
Vern Paulsen (Houston, TX, USA) Bernd Silbermann (Chemnitz, Germany)
Mihai Putinar (Santa Barbara, CA, USA) Harold Widom (Santa Cruz, CA, USA)
Ilya M. Spitkovsky (Williamsburg, VA, USA)

Subseries
Linear Operators and Linear Systems
Subseries editors:
Daniel Alpay (Beer Sheva, Israel)
Birgit Jacob (Wuppertal, Germany)
André C.M. Ran (Amsterdam, The Netherlands)

Subseries
Advances in Partial Differential Equations
Subseries editors:
Bert-Wolfgang Schulze (Potsdam, Germany)
Michael Demuth (Clausthal, Germany)
Jerome A. Goldstein (Memphis, TN, USA)
Nobuyuki Tose (Yokohama, Japan)
Ingo Witt (Göttingen, Germany)
Daniel Alpay • Bernd Kirstein
Editors
Recent Advances in Inverse
Scattering, Schur Analysis
and Stochastic Processes
A Collection of Papers Dedicated
to Lev Sakhnovich
Editors
Daniel Alpay Bernd Kirstein
Department of Mathematics Mathematisches Institut
Ben-Gurion University of the Negev Universität Leipzig
Beer Sheva, Israel Leipzig, Germany
ISSN 0255-0156 ISSN 2296-4878 (electronic)
Operator Theory: Advances and Applications
ISBN 978-3-319-10334-1 ISBN 978-3-319-10335-8 (eBook)
DOI 10.1007/978-3-319-10335-8
Library of Congress Control Number: 2015935221
Mathematics Subject Classification (2010): 47A57, 93C05, 60B20
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Contents
Editorial Introduction .................................................... vii
Part 1: Biographical Material and List of Publications of L.A. Sakhnovich
L.A. Sakhnovich – Biography ............................................ 3
List of Publications of L.A. Sakhnovich .................................. 9
L.A. Sakhnovich
My Teachers and Studies ........................................... 23
Part 2: Reserach Papers
D. Alpay, F. Colombo and I. Sabadini
Inner Product Spaces and Krein Spaces in the
Quaternionic Setting ................................................ 33
D. Alpay, P. Jorgensen, I. Lewkowicz and I. Martziano
Infinite Product Representations for Kernels and
Iterations of Functions .............................................. 67
Y. Arlinski˘ı and S. Hassi
Q-functions and Boundary Triplets of Nonnegative Operators ...... 89
S. Boiko, V. Dubovoy and A. Kheifets
On Some Special Cases of the Radon–Nikodym Theorem
for Vector- and Operator-valued Measures .......................... 131
A.E. Frazho, S. ter Horst and M.A. Kaashoek
State Space Formulas for a Suboptimal Rational Leech Problem II:
Parametrizationof All Solutions .................................... 149
vi Contents
B. Fritzsche, B. Kirstein and C. M¨adler
On a Simultaneous Approach to the Even and Odd Truncated
Matricial Hamburger Moment Problems ............................. 181
F. Gesztesy and R. Nichols
A Jost–Pais-type Reduction of (Modified) Fredholm Determinants
for Semi-separable Operators in Infinite Dimensions ................. 287
K.A. Makarov and E. Tsekanovski˘i
On the Addition and Multiplication Theorems ...................... 315
J. Rovnyak and L.A. Sakhnovich
On Indefinite Cases of Operator Identities Which Arise
in Interpolation Theory. II .......................................... 341
A. Sakhnovich and L. Sakhnovich
Nonlinear Fokker–PlanckEquation: Stability, Distance
and the Corresponding Extremal Problem in the Spatially
Inhomogeneous Case ................................................ 379
Editorial Introduction
Daniel Alpay and Bernd Kirstein
This volume is dedicated to Lev Aronovich Sakhnovich, on the occasion of his
80th birthday. Lev Aronovich is an outstanding expert in operator theory and
its applications, and his mathematical career is intimately related with the town
OdessaintheUkraine.wherehestudiedmathematicsatthePedagogicalInstitute.
V.P. Potapov, who was a Professorat the PedagogicalInstitute during that time,
very early observed the extraordinary abilities of the young student Sakhnovich.
The strong support of V.P. Potapov allowed Lev Aronovich to become a doctoral
student at the Pedagogical Institute in 1953. His advisor was M.S. Livsic, one of
the pioneers of operator theory. About three years later the main results of the
candidatethesiswerepresentedinthefamousseminarofMarkGrigorievichKrein
inOdessa.ThiscandidatethesiswasheldinextremelyhighesteembyM.G.Krein
who had the opinion that the thesis even deserved the second doctorate degree
(habilitation). The candidate thesis was the starting point of an extraordinary
scientific career, the main steps of which are reflected in the biographic material
which is contained in the first part of the volume. After the political changes in
the SovietUnionLev Aronovichwasallowedto takepartinconferencesinforeign
countriesandtowritemonographsinWesternpublishinghouses.Heisfamousfor
his far-reaching method of operator identities which turned out to be a universal
tool in several branches of analysis and stochastic processes (see in particular his
monograph [14] on Levy processes).
ItshouldbementionedthatLevAronovichwashonouredinaremarkableway
byLeipzigUniversity.IntheWintersemester2007/2008hewasawardedtheLeib-
nizguestprofessorshipofLeipzigUniversity.Uptonowthiswastheuniquetimein
the history of Leipzig University that a mathematician was honoured in this way.
The volume contains bibliographical material as well as a collection of ten
selected and refereed papers. The ten papers can be divided into four main (over-
lapping) families:
Interpolation and Moment problems: The work Infinite product representations
for kernels and iterations of functions, by Daniel Alpay, Palle Jorgensen, Izchak
LewkowiczandItzikMartziano,containsinparticularanexampleofalinearcom-
bination interpolation problem, where a linear combination of values at different
viii D. Alpay and B. Kirstein
nodes, is fixed, rather than the values at the nodes themselves. The work State
space formulas for a suboptimal rational Leech problem II: Parametrization of all
solutions,byA.E.Frazho,S.terHorstandM.A.Kaashoekisadirectcontinuation
ofthe paper [6] where the authorsdiscussedthe maximumentropysolutionofthe
interpolationproblem under consideration. It explicits connections between inter-
polation, Leech’s factorization theorem (see [9, 10]) and the state space method.
Next, the paperOn a simultaneous approach to the even and odd truncated matri-
cial Hamburger moment by Bernd Fritzsche, Bernd Kirstein and Conrad Ma¨dler,
continues the former investigations of the authors on matricial versions of power
moment problems (see [4, 5, 7] and the papers in the volume [1]). The approach
is based on Schur analysis, The main tool consists of an appropriate adaptation
of the classical algorithmdue to I. Schur and R. Nevanlinna to the moment prob-
lems under consideration. It should be mentioned that the truncated matricial
Hamburger matrix moment problems with an odd or even number of prescribed
moments will be handled in the most general case.
Aspects of indefinite inner product spaces: Here one can find the paper Inner
product spaces and Krein spaces in the quaternionic setting,byDanielAlpay,Fab-
rizio Colombo, and Irene Sabadini, which lays the foundations of quaternionic
Krein spaces (the Pontryagin space case had been considered in [2]). The paper
On indefinite cases of operator identities which arise in interpolation theory. IIby
J. Rovnyak and L.A. Sakhnovich, relates interpolation problems, operator identi-
ties, andKrein–LangerrepresentationofgeneralizedCarath´eodoryfunctions. The
authors studied the case of Nevanlinna functions in [11].
Operator-valued functions: Here we find the paper: Q-functions and boundary
triplets of non-negative operators, by Yu.M. Arlinskii and S. Hassi, where the no-
tion of Q-function is used in the setting of non-negative operators. In particular
a result of Krein and Ovˇcarenko (see [8]) is made more precise. The paper On
some special cases of the Radon–Nikodym theorem for vector- and operator-valued
measures, by S. Boiko,V. DubovoyandA. Kheifets, studies operator-valuedmea-
sures. Such measures play an important role in representation of operator-valued
functions appearing in operator theory. See for instance [3]. Also in this category
are the papers: A Jost–Pais-type reduction of (modified) Fredholm determinants
for semiseparable operators in infinite dimensions by Fritz Gesztesy and Roger
Nichols, and On the addition and multiplication theorems by K.A. Makarov and
E. Tsekanovskii.
Non linear differential equations: Non linear equations have always been an im-
portantresearchtopic for Lev Sakhnovich(see for instance [13, 12]) and it is very
fitting that the paper Nonlinear Fokker–Planck equation: stability, distance and
the corresponding extremal problem in the spatially inhomogeneous case, written
by Alexander Sakhnovich and Lev Sakhnovich, appears in the volume.
These various papers cover a wide range of the interests of Lev Sakhnovich, and
contain material which appears for the first time in print (as opposed to survey
papers).
Editorial Introduction ix
References
[1] D.Alpay and B. Kirstein (eds.): Interpolation, SchurFunctionsand Moment Prob-
lems II,Operator Theory: Advancesand Applications, Volume226, Springer,Basel
2012.
[2] D. Alpay and M. Shapiro. Reproducing kernel quaternionic Pontryagin spaces. In-
tegral Equations and Operator Theory, 50:431–476, 2004.
[3] M.S.Brodski˘ı. Triangular and Jordan representations of linear operators. American
Mathematical Society, Providence, R.I.,1971. Translated from theRussian byJ.M.
Danskin,Translations of Mathematical Monographs, vol. 32.
[4] Yu.M.Dyukarev,B.Fritzsche,B.Kirstein,andC.M¨adler,andH.C.Thiele.Ondis-
tinguished solutions of truncated matricial Hamburger moment problems. Complex
Analysis and Operator Theory 3(4):759–834, 2009.
[5] Yu.M. Dyukarev, B. Fritzsche, B. Kirstein, and C. M¨adler. On truncated matricial
Stieltjes typemoment problems. Complex Analysis and Operator Theory, 4(4):904–
951, 2010.
[6] A.E. Frazho, S. ter Horst, and M.A. Kaashoek. State space formulas for a subop-
timal rational Leech problem I: Maximum entropy solution. Integral Equations and
Operator Theory, 79:533–553, 2014.
[7] B.Fritzsche,B.Kirstein,andC.M¨adler.OnHankelnonnegativedefinitesequences.
thecanonicalHankelparametrization,andorthogonalpolynomials.Complexanalysis
and Operator Theory, 5(2):447–511, 2011.
[8] M.G. Kre˘ın and I.E. Ovˇcarenko. Inverse problems for Q-functions and resolvent
matrices of positive Hermitian operators. Dokl. Akad. Nauk SSSR, 242(3):521–524,
1978.
[9] R.B. Leech. Factorization of analytic functions and operator inequalities. Unpub-
lished manuscript.Available at:
http://www.people.virginia.edu/∼jlr5m/papers/leech.ps.
[10] R.B. Leech. Factorization of analytic functions and operator inequalities. Integral
Equations Operator Theory, 78(1):71–73, 2014.
[11] J. Rovnyak and L.A. Sakhnovich. On indefinite cases of operator identities which
arise in interpolation theory, volume 171 of Oper. Theory Adv. Appl., pages 281–
322. Birkha¨user, Basel, 2007.
[12] L.A. Sakhnovich. A hyperbolic sine-Gordon equation. Izv. Vyssh. Uchebn. Zaved.
Mat.,(1):54–63, 1991.
[13] L.A.Sakhnovich.Integrablenonlinearequationsonthesemi-axis.Ukrain. Mat. Zh.,
43(11):1578–1584, 1991.
[14] L.A. Sakhnovich. Levy Processes, Integral Equations, Statistical Physics: Connec-
tions and Interactions, volume225 of Operator Theory: Advances and Applications.
SpringerBasel, 2012.
Daniel Alpay Bernd Kirstein
Department of Mathematics Mathematisches Institut
Ben-Gurion Universityof theNegev Universit¨at Leipzig
P.O.B. 653 Augustusplatz 10/11
Beer-Sheva,Israel D-04109 Leipzig, Germany
e-mail: [email protected] e-mail: [email protected]