Operator Theory: Advances and Applications Vol. 210 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 (Vienna, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Honorary and Advisory Editorial Board: Vadim Adamyan (Odessa, Ukraine) Lewis A. Coburn (Buffalo, NY, USA) Albrecht Böttcher (Chemnitz, Germany) Ciprian Foias (College Station, TX, USA) B. Malcolm Brown (Cardiff, UK) J. William Helton (San Diego, CA, USA) Raul Curto (Iowa City, IA, USA) Thomas Kailath (Stanford, CA, USA) Fritz Gesztesy (Columbia, MO, USA) Peter Lancaster (Calgary, AB, Canada) Pavel Kurasov (Lund, Sweden) Peter D. Lax (New York, NY, USA) Leonid E. Lerer (Haifa, Israel) 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) Leiba Rodman (Williamsburg, VI, USA) Ilya M. Spitkovsky (Williamsburg, VI, 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) 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 Apartado Postal14-740 Mexico,D.F. Mexico e-mail: [email protected] 2010 Mathematics Subject Classification: 47-06 Library of Congress Control Number: 2010928340 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de ISBN 978-3-0346-0547-2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2010 Springer Basel AG P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-0346-0547-2 e-ISBN 978-3-0346-0548-9 9 8 7 6 5 4 3 2 1 www.birkhauser.ch 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