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) Tel Aviv University B. Mityagin (Columbus) Ramat Aviv, Israel V. Olshevsky (Storrs) M. Putinar (Santa Barbara) Editorial Board: L. Rodman (Williamsburg) D. Alpay (Beer-Sheva) J. Rovnyak (Charlottesville) J. Arazy (Haifa) D. E. Sarason (Berkeley) A. Atzmon (Tel Aviv) I. M. Spitkovsky (Williamsburg) J. A. Ball (Blacksburg) S. Treil (Providence) A. Ben-Artzi (Tel Aviv) H. Upmeier (Marburg) H. Bercovici (Bloomington) S. M. Verduyn Lunel (Leiden) A. Böttcher (Chemnitz) D. Voiculescu (Berkeley) K. Clancey (Athens, USA) D. Xia (Nashville) L. A. Coburn (Buffalo) D. Yafaev (Rennes) R. E. Curto (Iowa City) K. R. Davidson (Waterloo, Ontario) Honorary and Advisory R. G. Douglas (College Station) Editorial Board: A. Dijksma (Groningen) C. Foias (Bloomington) H. Dym (Rehovot) P. R. Halmos (Santa Clara) P. A. 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For any kind of use permission of the copyright owner must be obtained. © 2007 Birkhäuser Verlag, 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 (cid:102) Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN-10: 3-7643-7736-4 e-ISBN-10: 3-7643-7737-2 ISBN-13: 978-3-7643-7736-6 e-ISBN-13: 978-3-7643-7737-3 9 8 7 6 5 4 3 2 1 www.birkhauser.ch 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.