Modern Operator Theory and Applications

259 Pages·2006·2.614 MB·English

by  Yakob M. Erusalimskii, Yakob M. Erusalimsky;Israel Gohberg;Sergei M. Grudsky;Vladimir Rabinovich;Nicolai Vasilevski

Operator Theory: Advances and Applications
Vol. 170
Editor:
I. Gohberg
Editorial Office: S. T. Kuroda (Tokyo)
School of Mathematical P. Lancaster (Calgary)
Sciences L. E. Lerer (Haifa)
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Editorial Board:
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Subseries: Advances in
Michael Demuth
Partial Differential Equations
Technische Universität Clausthal
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Universität Potsdam The University of Memphis, TN
Germany USA
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Keio University, Yokohama
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Japan
Modern Operator Theory
and Applications
The Igor Borisovich Simonenko Anniversary Volume
Ya. M. Erusalimsky
I. Gohberg
S. M. Grudsky
V. Rabinovich
N. Vasilevski
Editors
Birkhäuser Verlag
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Basel Boston Berlin
Editors:
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Contents
Portrait of Igor Borisovich Simonenko .................................... vii
Introduction
Ja.M. Jerusalimsky
Life and Work of Igor BorisovichSimonenko ......................... 1
V.S. Pilidi
Operators of Local Type and Singular Integral Operators ............ 2
V.B. Levenshtam, S.M. Zenkovskaya
An Averaging Method and its Application
to Hydrodynamics Problems ......................................... 5
List of Ph.D., whose supervisor was I.B. Simonenko ....................... 12
List of Ph.D., whose co-supervisor was I.B. Simonenko .................... 13
List of D.Sc., whose advisor was I.B. Simonenko .......................... 13
Principal Publications of I.B. Simonenko .................................. 13
Contributions
A.B. Antonevich
Coefficients Averaging for Functional Operators
Generated by Irrational Rotation .................................... 27
A. B¨ottcher and D. Wenzel
On the Verification of Linear Equations and the Identification
of the Toeplitz-plus-Hankel Structure ................................ 43
L.P. Castro, R. Duduchava and F.-O. Speck
Asymmetric Factorizations of Matrix Functions on the Real Line ..... 53
R.G. Douglas and C. Foias
On the Structure of the Square of a C (1) Operator .................. 75
0
I. Feldman, N. Krupnik and A. Markus
On the Connection Between the Indices of a Block Operator Matrix
and of its Determinant ............................................... 85
vi Contents
I. Gohberg, M.A. Kaashoek and L. Lerer
Quasi-commutativity of Entire Matrix Functions and
the Continuous Analogue of the Resultant ........................... 101
S.M. Grudsky
Double Barrier Options Under L´evy Processes ....................... 107
Yu.I. Karlovich
A Local-trajectory Method and Isomorphism Theorems
for Nonlocal C∗-algebras ............................................ 137
V. Kokilashvili, V. Paatashvili and S. Samko
Boundedness in Lebesgue Spaces with Variable Exponent of
the Cauchy Singular Operator on Carleson Curves ................... 167
V.B. Levenshtam
On the Averaging Method for the Problem of Heat Convection
in the Field of Highly-Oscillating Forces ............................. 187
V.S. Rabinovich, S. Roch and B. Silbermann
Finite Sections of Band-dominated Operators with
Almost Periodic Coefficients ......................................... 205
N. Vasilevski
On the Toeplitz Operators with Piecewise Continuous Symbols
on the Bergman Space .............................................. 229
H. Widom
Asymptotics of a Class of Operator Determinants .................... 249
Igor Borisovich Simonenko
OperatorTheory:
Advances andApplications,Vol.170,1–26
(cid:1)c 2006Birkh¨auserVerlagBasel/Switzerland
Introduction
Ja.M. Jerusalimsky
Life and Work of Igor Borisovich Simonenko
In August of 2005,the eminent Russian mathematicianDr. Igor BorisovichSimo-
nenko celebrated his 70th Birthday.
IgorBorisovichwasborninKiev(Ukraine,formerUSSR),wherehespenthis
childhood. Along with the majority of his contemporaries, he experienced all the
difficultiesofwartime,evacuationandoccupation,togetherwithhismotherinthe
steppes of Salsk. In 1943, upon returning to Lugansk with his mother, he began
school, entering the third grade. In 1947 he left the primary school and entered
a machine-building technical school. Having graduated from school in 1953, Igor
firstworkedinafactoryandthenbegantostudy atthePhysicsandMathematics
Department of the Rostov State University.
The greatestinfluence onthe youngmathematicianI.B. Simonenkowas ren-
dered by his teacher and supervisor, the brilliant scientist Fyodor Dmitrievich
Gakhov,who managedto createthree scientificschools:inKazan,Rostov-on-Don
and Minsk.
InIgorBorisovich’sstudentyears,thePhysicsandMathematicsDepartment
of Rostov State University was on the rise. This had much to do with the pres-
ence of the talented young experts in mechanics, the Moscow State University
graduates I.I. Vorovich,N.N. Moiseev and L.A. Tolokonnikov (later academicians
of the Russian Academy of Science) and the arrival at RSU in 1953 of professor
F.D. Gakhov from Kazan.
An active influence on the scientific life of the department was rendered
by the scientific seminar “Boundary value problems” (headed by F.D. Gakhov)
and the seminar “Theory of nonlinear operators” (headed by I.I. Vorovich and
M.G. Khaplanov). The latter seminar became the source of ideas and methods in
functional analysis and the starting point of a wide range of application of these
methods by the Rostov mathematicians.
Inhis1961Ph.D.thesis“Treatiesinthetheoryofsingularintegraloperators”
I.B. Simonenko followed the classical methods of the school of his teacher. After
defending this thesis, I.B. worked for several years at the RSU computer center.
During this period, the results on the problems of electrostatics were obtained
2 Introduction
(jointly with V.P. Zakharyuta and V.I. Yudovich), including a calculation of the
capacity of condensers of complex form and dielectric materials with complex
structure.
In1967,attheageof32,sixyearsafterhe defendedhisPh.D.thesis,I.B.Si-
monenko defended his thesis for a degree of Doctor of Science. In this thesis,
entitled “Operators of local type and some other problems of the theory of linear
operators,” he sharply turned towards the wide usage of the general methods of
functional analysis.In 1971 professorI.B. Simonenko became the head of the Nu-
merical Mathematics Chair. The following year this chair was split into two; I.B.
became the head of one of them, the Chair of Algebra and Discrete Mathematics.
The Chair of Algebra and Discrete Mathematics can be rightfully called the
Chair of I.B. Simonenko. Here he worked together with his colleagues, students,
and the students of his students. Here he fully developed his teaching talent. He
lectured on “Algebra and geometry,” “Mathematical logic,” “Discrete mathemat-
ics,” and “Mathematical analysis”.
The scientific seminar of the Chair of Algebra and Discrete Mathematics
is widely known both in Russia and abroad. Besides I.B. and his students, such
well-known mathematicians as
S.G. Mikhlin, I.Tz. Gokhberg, N.Ya. Krupnik, B.A. Plamenevsky,
P.E. Sobolevsky, A.I. Volpert, A.S. Markus, A.P. Soldatov,
R.V. Duduchava, A.S. Dynin, B. Silbermann, A. Bo¨ttcher,
M.V. Fedoryuk, G.S. Litvinchuk, I.M. Spitkovsky, N.L. Vasilevski,
A.B. Antonevich, N.N. Vragov, Yu.I. Karlovich, S.G. Samko,
N.K. Karapetiantz
and others gave talks here.
Whileareputedscientistandtheheadofwidelyknownscientificschool,Igor
Borisovichremainsamodestandcharmingman.Ifaskedtodescribehiminseveral
words, I would leave only two – the Scientist and the Teacher.
V.S. Pilidi
Operators of Local Type and Singular Integral Operators
We recall the main definitions from the theory of Fredholm operators. Let X be
a Banach space. Denote by B(X) (K(X)) the set of all linear continuous (all com-
pact) operators acting on the space X. An operator A ∈ B(X) is called Fredholm
(Φ-operator) if its kernel is finite dimensional and the range is closed and has fi-
nite codimension.1. The Fredholm property of A is equivalent to the existence of
operators R , R ∈ B(X) such that the following equalities hold: R A = I +T ,
1 2 1 1
AR =I+T , where T , T ∈K(X). The operatorsR and R are called left and
2 2 1 2 1 2
rightregularizorsofA. The existence of regularizorsis evidently equivalentto the
1The two terms mentioned have practically superseded the earlier term “Noether operator,”
whichwasusedbyI.B.Simonenkoinhisclassicpaper “Thenewgeneralmethod...”
Introduction 3
invertibilityof theresidueclassA+K(X)inthe quotientalgebraB(X)/K(X)(the
“Calkin algebra”).Let us call operators A and B equivalent if B−A∈K(X). We
note the following trivial fact: if two operators are equivalent, then the Fredholm
property of one of them implies this property for the other.
ClassicalGelfand theory is in some sense a local principle, giving in the case
of commutative Banach algebra conditions for invertibility in some “local” terms.
I.B. Simonenko’s local method is, in essence, an analogue of this theory2.
Speakinginalgebraicterms,thisprinciplepermitstoobtaincriteriaofinvertibility
of elements of Calkin algebras in noncommutative case.
We explain the definition of an operator of local type, given below, with the
following example. Consider the singular integral operator
(cid:1)
1 1 f(y)
(Sf)(x)= dy
πi y−x
0
acting on the space L (0,1), where the integral is understood in the sense of
2
principal value. Let P be the operator of multiplication by the characteristic
F
function of the measurable set F ⊂ [0,1] acting on the same space. If F and F
1 2
areclosednonintersecting subsets of the segment[0,1],then the integraloperator
P SP hasboundedkernel,andthereforeiscompact.Notethatthecompactness
F1 F2
of this operator is related to the fact that the strong singularity of the kernel lies
on the diagonal of its domain of definition.
Now let us pass to the general definition of operator of local type. Let X
be a compact Hausdorff space. Suppose that a σ-finite nonnegative measure is
definedonthisspace,suchthatallopensubsetsofX aremeasurable.Anoperator
A∈B(L (X))(1≤p<∞)is calledanoperatorof local type ifforanytwoclosed
p
disjoint subsets F , F ⊂ X, the operator P AP is compact. This definition is
1 2 F1 F2
equivalent to the following: for any continuous function ϕ on X, the commutator
ϕA−AϕI is compact.
InthesequelwewillsupposethatthespaceX andthenumberparefixedand
that all operators under consideration are operators of local type. The notation
K(L (X)) will be shortened to K.
p
Operators A and B are called locally equivalent at the point x ∈ X when
inf|(B−A)P |=0,where|·|denotestheseminormmodulothesetofallcompact
u
u
operators,andthe greatestlowerbound is takenoverthe setof allneighborhoods
of x in X (this notion will be expressed as A∼x B).
An operator A is called locally Fredholm at the point x ∈ X if there exist
operators R , R such that R A∼x I, AR ∼x I.
1 2 1 2
Themainassertionofthelocalprincipleisasfollows:anoperator isFredholm
if and only if it is locally Fredholm at every point of X.
The following statement plays an essential role: if two operators are locally
equivalent at some point, then the local Fredholm property for one of them implies
2The idea of this comparison is mentioned in the book R. Hagen, S. Roch, B. Silbermann,
C∗-Algebras and Numerical Analysis,2001,p.204.