Recent Trends in Toeplitz and Pseudodifferential Operators: The Nikolai Vasilevskii Anniversary Volume

274 Pages·2010·4.24 MB·English

by  Sergei Grudsky

Operator Theory: Advances and Applications
Vol. 210
Founded in 1979 by Israel Gohberg
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Recent Trends in
Toeplitz and
Pseudodifferential
Operators
The Nikolai Vasilevskii Anniversary Volume
Roland Duduchava
Israel Gohberg
Sergei M. Grudsky
Vladimir Rabinovich
Editors
Birkhäuser
Editors:
Roland Duduchava Israel Gohberg (Z”L)
Department of Mathematical Physics
Andrea Razmadze Mathematical Institute Vladimir Rabinovich
M. Alexidze str. 1 National Polytechnic Institute
Tbilisi 0193 ESIME Zacatenco
Georgia Av. IPN
e-mail: [email protected] 07738 Mexico, D.F.
Mexico
Sergei M. Grudsky e-mail:[email protected]
Departamento de Matemáticas
CINVESTAV
Av. I.P.N. 2508
Col. San Pedro Zacatenco
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Mexico
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Contents
S. Grudsky, Y. Latushkin and M. Shapiro
The Life and Work of Nikolai Vasilevski ............................. 1
Principal publications of Nikolai Vasilevski ........................... 6
A. B¨ottcher, S.M. Grudsky an E.A. Maksimenko
On the Structure of the Eigenvectors of Large Hermitian
Toeplitz Band Matrices .............................................. 15
V.D. Didenko
Complete Quasi-wanderingSets and Kernels
of Functional Operators ............................................. 37
R. Duduchava
Lions’ Lemma, Korn’s Inequalities and the Lam´e Operator
on Hypersurfaces .................................................... 43
J.O. Gonza´lez-Cervanves, M.E. Luna-Elizarrara´s and M. Shapiro
On the Bergman Theory for Solenoidal and Irrotational
Vector Fields, I: General Theory ..................................... 79
M.G. Hajibayov and S.G. Samko
Weighted Estimates of Generalized Potentials in Variable
Exponent Lebesgue Spaces on Homogeneous Spaces .................. 107
Yu.I. Karlovich and J. Loreto Hern´andez
Wiener-Hopf Operators with Oscillating Symbols
on Weighted Lebesgue Spaces ........................................ 123
V.V. Kucherenko and A. Kryvko
On the Ill-posed Hyperbolic Systems with a Multiplicity
Change Point of Not Less Than the Third Order ..................... 147
M. Loaiza and A. S´anchez-Nungaray
On C∗-Algebras of Super Toeplitz Operators with Radial Symbols ... 175
vi Contents
H. Mascarenhas and B. Silbermann
Universality of Some C∗-Algebra Generated by a Unitary
and a Self-adjoint Idempotent ....................................... 189
R. Quiroga-Barranco
Commutative Algebras of Toeplitz Operators and
LagrangianFoliations ................................................ 195
V. Rabinovich and S. Roch
Exponential Estimates of Eigenfunctions of Matrix
Schro¨dinger and Dirac Operators .................................... 203
N. Tarkhanov
The Laplace-BeltramiOperator on a Rotationally
Symmetric Surface ................................................... 217
A. Unterberger
On the Structure of Operators with Automorphic Symbols ........... 233
H. Upmeier
Hilbert Bundles and Flat Connexions over Hermitian
Symmetric Domains ................................................. 249
Nikolai Vasilevski
OperatorTheory:
Advances andApplications,Vol.210,1–14
(cid:2)c 2010SpringerBaselAG
The Life and Work of Nikolai Vasilevski
Sergei Grudsky, Yuri Latushkin and Michael Shapiro
Nikolai Leonidovich Vasilevski was born on January 21, 1948 in Odessa,Ukraine.
His father, Leonid Semenovich Vasilevski, was a lecturer at Odessa Institute of
Civil Engineering, his mother, Maria Nikolaevna Krivtsova, was a docent at the
Department of Mathematics and Mechanics of Odessa State University.
In 1966 Nikolai graduated from Odessa High School Number 116, a school
with special emphasis in mathematics and physics, that made a big impact at his
creativeandactiveattitudenotonly tomathematics,but tolife ingeneral.Itwas
a very selective high school accepting talented children from all over the city, and
famousforahighqualityselectionofteachers.Acreative,nonstandard,andatthe
same time highly personal approach to teaching was combined at the school with
a demanding attitude towards students. His mathematics instructor at the high
school was Tatjana Aleksandrovna Shevchenko, a talented and dedicated teacher.
The school was also famous because of its quite unusual by Soviet standards sys-
tem of self-government by the students. Quite a few graduates of the school later
became well-known scientists, and really creative researchers.
In 1966 Nikolai became a student at the Department of Mathematics and
Mechanics of Odessa State University. Already at the third year of studies, he
began his serious mathematical work under the supervision of the well-known
Soviet mathematician Georgiy Semenovich Litvinchuk. Litvinchuk was a gifted
teacher and scientific adviser. He, as anyone else, was capable of fascinating his
studentsbynewproblemswhichhavebeenalwaysinterestingandup-to-date.The
weekly Odessa seminar on boundary value problems, chaired by Prof. Litvinchuk
for more than 25 years, very much influenced Nikolai Vasilevski as well as others
students of G.S. Litvinchuk.
N.VasilevskistartedtoworkontheproblemofdevelopingtheFredholmthe-
oryforaclassofintegraloperatorswithnonintegrableintegralkernels.Inessence,
the integral kernel was the Cauchy kernel multiplied by a logarithmic factor. The
integral operators of this type lie between the singular integral operators and the
integral operators whose kernels have weak (integrable) singularities. A famous
Soviet mathematician F.D. Gakhov posted this problem in early 1950ies, and it
remainedopenformorethan20years.Nikolaimanagedto provideacomplete so-
lution in the setting which was much more general than the original. Working on
2 S. Grudsky, Y. Latushkin and M. Shapiro
thisproblem,Nikolaihasdemonstratedoneofthemaintraitsofhismathematical
talent: his ability to achieve a deep penetration in the core of the problem, and
to see rather unexpected connections between different theories. For instance, in
order to solve Gakhov’s Problem, Nikolai utilized the theory of singular integral
operators with coefficients having discontinuities of first kind, and the theory of
operators whose integral kernels have fixed singularities – both theories just ap-
pearedat that time. The success of the young mathematician was wellrecognized
by a broad circle of experts working in the area of boundary value problems and
operatortheory. In 1971Nikolai was awardedthe prestigious M. OstrovskiiPrize,
giventotheyoungUkrainianscientistsforthebestresearchwork.Duetohissolu-
tionofthefamousproblem,Nikolaiquicklyenteredthe mathematicalcommunity,
andbecameknowntomanyprominentmathematiciansofthattime.Inparticular,
hewasverymuchinfluencedbythe hisregularinteractionswithsuchoutstanding
mathematicians as M.G. Krein and S.G. Mikhlin.
In1973N.VasilevskidefendedhisPhDthesisentitled“TotheNoethertheory
ofaclassofintegraloperatorswithpolar-logarithmickernels”.Inthesameyearhe
became anAssistantProfessoratthe Department ofMathematica andMechanics
of Odessa State University, where he was later promoted to the rank of Associate
Professor,and, in 1989, to the rank of Full Professor.
Having receivedthe degree,Nikolai continuedhis active mathematicalwork.
Soon, he displayed yet another side of his talent in approaching mathematical
problems: his vision and ability to use general algebraic structures in operator
theory, which, on one side, simplify the problem, and, on another, can be used in
many other problems. We will briefly describe two examples of this.
The first example is the method of orthogonal projections. In 1979, study-
ing the algebra of operators generated by the Bergman projection, and by the
operators of multiplication by piece-wise continuous functions, N. Vasilevski gave
a description of the C∗-algebra generated by two self-adjoint elements s and n
satisfying the properties s2 + n2 = e and sn + ns = 0. A simple substitution
p=(e+s−n)/2 and q =(e−s−n)/2 shows that this algebra is also generated
bytwoself-adjointidempotents(orthogonalprojections)pandq (andthe identity
elemente).Duringthelastquarterofthepastcentury,thelatteralgebrahasbeen
rediscovered by many authors all over the world. Among all algebras generated
by orthogonal projections, the algebra generated by two projections is the only
tame algebra (excluding the trivial case of the algebra with identity generated by
one orthogonal projection). All algebras generated by three or more orthogonal
projections are known to be wild, even when the projections satisfy some addi-
tional constrains. Many model algebras arising in operator theory are generated
by orthogonal projections, and thus any information of their structure essentially
broadens the set of operator algebras admitting a reasonable description. In par-
ticular, two and more orthogonal projections naturally appear in the study of
various algebras generated by the Bergman projection and by piece-wise contin-
uous functions having two or more different limiting values at a point. Although
theseprojections,say,P,Q ,...,Q ,satisfyanextraconditionQ +···+Q =I,
1 n 1 n
The Life and Work of Nikolai Vasilevski 3
they still generate, in general, a wild C∗-algebra.At the same time, it was shown
that the structure of the algebra just mentioned is determined by the joint prop-
erties of(cid:2)certain positive injective contractions Ck, k = 1,...,n, satisfying the
identity n C =I,and,therefore,thestructureisdeterminedbythestructure
k=1 k
of the C∗-algebragenerated by the contractions. The principal difference between
the case of two projections and the general case of a finite set of projections is
now completely clear: for n = 2 (and the projections P and Q+(I −Q) = I)
we have only one contraction, and the spectral theorem directly leads to the de-
sired description of the algebra. For n ≥ 2 we have to deal with the C∗-algebra
generatedby afinite setofnoncommutingpositiveinjective contractions,whichis
a wild problem. Fortunately, for many important cases related to concrete oper-
ator algebras, these projections have yet another special property: the operators
PQ P,...,PQ P mutuallycommute.Thispropertymakesthe respectivealgebra
1 n
tame, and thus it has a nice and simple description as the algebra of all n×n
matrix-valued functions that are continuous on the joint spectrum Δ of the oper-
ators PQ P,...,PQ P, and have certain degeneration on the boundary of Δ.
1 n
Another notable example of the algebraic structures used and developed by
N. Vasilevski is his version of the Local Principle. The notion of locally equiva-
lent operators, and localization theory were introduced and developed by I. Si-
monenko in mid-sixtieth. According to the tradition of that time, the theory was
focused on the study of individual operators, and on the reduction of the Fred-
holmpropertiesofanoperatortolocalinvertibility.Later,differentversionsofthe
local principle have been elaborated by many authors, including, among others,
G.R. Allan, R. Douglas, I.Ts. Gohberg and N.Ia. Krupnik, A. Kozak, B. Silber-
mann. In spite of the fact that many of these versions are formulated in terms of
Banach- or C∗-algebras, the main result, as before, reduces invertibility (or the
Fredholm property) to local invertibility. On the other hand, at about the same
time,severalpapersonthedescriptionofalgebrasandringsintermsofcontinuous
sectionswerepublishedby J.Dauns andK.H.Hofmann,M.J.Dupr´e,J.M.G.Fell,
M. Takesaki and J. Tomiyama. These two directions have been developed inde-
pendently, with no known links between the two series of papers. N. Vasilevski
was the one who proposed a local principle which gives the global description of
the algebra under study in terms of continuous sections of a certain canonically
defined C∗-bundle. This approach is based on general constructions of J. Dauns
and K.H. Hofmann, and results of J. Varela. The main contribution consists of
a deep re-comprehension of the traditional approach to the local principles uni-
fying the ideas coming from both directions mentioned above, which results in a
canonicalprocedurethatprovidesthe globaldescriptionofthe algebraunder con-
siderationinterms ofcontinuoussections ofaC∗-bundle constructedby means of
local algebras.
In the eighties and even later, the main direction of the work of Nikolai
Vasilevskihasbeenthestudyofmulti-dimensionalsingularintegraloperatorswith
discontinuous coefficients. The main philosophy here was to study first algebras