Abstract: 1 The operator of singular integration.- 1.1 Notations, definitions and auxiliary statements.- 1.1.1 The operator of singular integration.- 1.1.2 The space Lp(?,?).- 1.1.3 Interpolation theorems.- 1.2 The boundedness of the operator S? in the space Lp(?) with ? being a simple curve.- 1.3 Nonsimple curves.- 1.4 Integral operators in weighted Lp spaces.- 1.5 Unbounded curves.- 1.6 The operator of singular integration in spaces of Holder continuous functions.- 1.7 The operator S?*.- 1.8 Exercises.- Comments and references.- 2 One-sided invertible operators.- 2.1 Direct sum of subspaces.- 2.2 The direct complement.- 2.3 Linear operators. Notations and simplest classes.- 2.4 Projectors connected with the operator of singular integration.- 2.5 One-sided invertible operators.- 2.6 Singular integral operators and related operators.- 2.7 Examples of one-sided invertible singular integral operators.- 2.8 Two lemmas on the spectrum of an element in a subalgebra of a Banach algebra.- 2.9 Subalgebras of a Banach algebra generated by one element.- 2.10 Exercises.- Comments and references.- 3 Singular integral operators with continuous coefficients.- 3.1 The index of a continuous function.- 3.2 Singular integral operators with rational coefficients.- 3.3 Factorization of functions.- 3.4 The canonical factorization in a commutative Banach algebra.- 3.5 Proof of the factorization theorem.- 3.6 The local factorization principle.- 3.7 Operators with continuous coefficients.- 3.8 Approximate solutions of singular integral equations.- 3.9 Generalized factorizations of continuous functions.- 3.10 Operators with continuous coefficients (continuation).- 3.11 Additional facts and generalizations.- 3.12 Operators with degenerating coefficients.- 3.13 A generalization of singular integral operators with continuous coefficients.- 3.14 Solution of Wiener-Hopf equations.- 3.15 Some applications.- 3.16 Exercises.- Comments and references.- 4 Fredholm operators.- 4.1 Normally solvable operators.- 4.2 The restriction of normally solvable operators.- 4.3 Perturbation of normally solvable operators.- 4.4 The normal solvability of the adjoint operator.- 4.5 Generalized invertible operators.- 4.6 Fredholm operators.- 4.7 Regularization of operators. Applications to singular integral operators.- 4.8 Index and trace.- 4.9 Functions of Fredholm operators and their index.- 4.10 The structure of the set of Fredholm operators.- 4.11 The Dependence of kerX and imX on the operator X.- 4.12 The continuity of the function kx.- 4.13 The case of a Hilbert space.- 4.14 The normal solvability of multiplication by a matrix function.- 4.15 ?+--operators.- 4.16 One-sided regularization of operators.- 4.17 Projections of invertible operators.- 4.18 Exercises.- Comments and references.- 5 Local Principles and their first applications.- 5.1 Localizing classes.- 5.2 Multipliers on $$ \mathop l\limits^ \sim _p $$.- 5.3 paired equations with continuous coefficients on $$ \mathop l\limits^ \sim _p $$.- 5.4 Operators of local type.- 5.5 Exercises.- Comments and references.- References.

Abstract: 1. Two-sided estimates for polynomials related to Newton's polygon and their application to studying local properties of partial differential operators in two variables.- 1. Newton's polygon of a polynomial in two variables.- 2. Polynomials admitting of two-sided estimates.- 3. N Quasi-elliptic polynomials in two variables.- 4. N Quasi-elliptic differential operators.- Appendix to 4.- 2. Parabolic operators associated with Newton's polygon.- 1. Polynomials correct in Petrovski?'s sense.- 2. Two-sided estimates for polynomials in two variables satisfying Petrovski?'s condition. N-parabolic polynomials.- 3. Cauchy's problem for N-stable correct and N-parabolic differential operators in the case of one spatial variable.- 4. Stable-correct and parabolic polynomials in several variables.- 5. Cauchy's problem for stable-correct differential operators with variable coefficients.- 3. Dominantly correct operators.- 1. Strictly hyperbolic operators.- 2. Dominantly correct polynomials in two variables.- 3. Dominantly correct differential operators with variable coefficients (the case of two variables).- 4. Dominantly correct polynomials and the corresponding differential operators (the case of several spatial variables).- 4. Operators of principal type associated with Newton's polygon.- 1. Introduction. Operators of principal and quasi-principal type.- 2. Polynomials of N-principal type.- 3. The main L2 estimate for operators of N-principal type.- Appendix to 3.- 4. Local solvability of differential operators of N-principal type.- Appendix to 4.- 5. Two-sided estimates in several variables relating to Newton's polyhedra.- 1. Estimates for polynomials in ?n relating to Newton's polyhedra.- 2. Two-sided estimates in some regions in ?n relating to Newton's polyhedron. Special classes of polynomials and differential operators in several variables.- 6. Operators of principal type associated with Newton's polyhedron.- 1. Polynomials of N-principal type.- 2. Estimates for polynomials of N-principal type in regions of special form.- 3. The covering of ?n by special regions associated with Newton's polyhedron.- 4. Differential operators of ?n-principal type with variable coefficients.- Appendix to 4.- 7. The method of energy estimates in Cauchy's problem 1. Introduction. The functional scheme of the proof of the solvability of Cauchy's problem.- 2. Sufficient conditions for the existence of energy estimates.- 3. An analysis of conditions for the existence of energy estimates.- 4. Cauchy's problem for dominantly correct differential operators.- References.

Abstract: In this paper we consider Toeplitz and Hankel operators on the Bergman spaces of the unit ball and the polydisk in C n whose symbols are bounded measurable functions. We giv e necessary and sufficient conditions on the symbols for these operators to be compact. We study the Fredholm theory of Topelitz operators for which the corresponding Hankel operator is compact. For these Toeplitz operators the essential spectrum is computed and shown to be connected. We also consider symbols that extend to continuous functions on the maximal ideal space of H ∞(Ω); for these symbols we describe when the Toeplitz or Hankel operators are compact

Abstract: For a general spectral operator, the author establishes types of algebraic structures of the spaces of the corresponding isospectral Lax operators, which essentially form the theoretical basis of the Lax operator method. Furthermore the author introduces the concepts of tau -algebras and master algebras to describe time-polynomial-dependent symmetries of nonlinear integrable equations. Finally the author applies the theory of Lax operators to the KP hierarchy of integrable equations as an illustrative example, and thus obtain the master symmetry algebra of the KP hierarchy.

Abstract: In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC *-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA 2(Ω) for a wide class of plane domains Ω⊂C, and in Fock spacesA 2(C N),N≧1.

Abstract: The symbols of hyponormal Toeplitz operators are completely described and those are also studied, being related with the extremal problems of Hardy spaces. Moreover, we discuss Halmos's question about a subnormal Toeplitz operator when the self-commutator is finite rank

Abstract: In this paper, we study some spectral properties of the discrete Schrodinger operator = Δ + q defined on a locally finite connected graph with an automorphism group whose orbit space is a finite graph. The discrete Laplacian and its generalization have been explored from many different viewpoints (for instance, see [2] [4]). Our paper discusses the discrete analogue of the results on the bottom of the spectrum established by T. Kobayashi, K. Ono and T. Sunada [3] in the Riemannian-manifold-setting.

Abstract: Foundations Perturbation theory Weakly decomposable operators and automatic continuity Invariant subspaces for subdecomposable operators Multivariate theory.

Abstract: This paper is a continuation of Miyake [7] by the first named author. We shall study the unique solvability of an integro-differential equation in the category of formal or convergent power series with Gevrey estimate for the coefficients, and our results give some analogue in partial differential equations to Ramis [10, 11] in ordinary differential equations. In the study of analytic ordinary differential equations, the notion of irregularity was first introduced by Malgrange [3] as a difference of indices of a differential operator in the categories of formal power series and convergent power series. After that, Ramis extended his theory to the category of formal or convergent power series with Gevrey estimate for the coefficients. In these studies, Ramis revealed a significant meaning of a Newton polygon associated with a differential operator.

Abstract: © Société mathématique de France, 1992, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: In this paper we study Hankel and Toeplitz operators on Dirichlet type spaces Dα. We obtain necessary and sufficient condition on the symbols for these operators to be bounded and to belong to the Schatten ideal Sp for certain α and p.

Abstract: Partial differential operators of elliptic type The Laplacian n Euclidean spaces Constructions and estimates of elementary solutions Smoothness of solutions Vishik-Sobolev problems General boundary value problems Schauder estimates and applications Degenerate elliptic operators.

Abstract: In this paper we consider operators acting on Clifford algebra valued polynomials and, in particular, differential operators with polynomial coefficients. The decomposition of polynomials into homogeneous pieces leads to the classical homogeneous decomposition of operators and the further decomposition of homogeneous polynomials into monogenic polynomials leads to the concept of monogenic operator. Monogenic operators are characterized in terms of commutation relations and the monogenic decomposition of differential operators is studied in detail.

Abstract: © Foundation Compositio Mathematica, 1992, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: We consider singular integral operators of the form (a)Z 1L−1Z2, (b)Z 1Z2L−1, and (c)L −1Z1Z2, whereZ 1 andZ 2 are nonzero right-invariant vector fields, andL is theL 2-closure of a canonical Laplacian. The operators (a) are shown to be bounded onL p for allp∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type (p, p) for anyp∈[1, ∞).

Abstract: We extend the results of our previous paper "C*-algebras and numerical linear algebra" to cover the case of "unilateral" sections. This situation bears a close resemblance to the case of Toeplitz operators on Hardy spaces, in spite of the fact that the operators here are far from Toeplitz operators. In particular, there is a short exact sequence 0 --> K --> A --> B --> 0 whose properties are essential to the problem of computing the spectra of self adjoint operators.

Abstract: Commutation properties of two‐dimensional momentum operators with gauge potentials are investigated. A notion of local quantization of magnetic flux is introduced to characterize physically the strong commutativity of the momentum operators. In terms of the notion, a necessary and sufficient condition is given for the position and the momentum operators to be equivalent to the Schrodinger representation of the canonical commutation relations.

Abstract: CONTENTSIntroduction §1. Technical preliminaries1. Symplectic and contact geometry2. Integral calculus on 3. Homogeneous and formally homogeneous generalized functions on §2. Fourier transforms of homogeneous functions1. Statement of the problem2. The main lemma3. Structure theorem for the Fourier transform of a homogeneous function4. Commutation formulae and the choice of constants §3. Fourier-Maslov integral operators1. The Maslov canonical operator2. Fourier-Maslov integral operators §4. Examples and applications1. Preliminary remarks2. The discontinuity propagation problem3. The discontinuity metamorphosis problem4. Investigation of Green's function of the Cauchy problem §5. Microlocal classification of pseudodifferential operators1. Microlocal equivalence2. Elliptic operators and operators of principal type3. Operators of subprincipal type §6. Equations of principal and subprincipal types1. Equations of principal type2. Proof of Theorem 13. Equations of subprincipal typeReferences

Abstract: Publisher Summary This chapter discusses the magnetic Schrodinger operator. The chapter provides a detailed account for several recent improvements of some results of Helffer. The chapter reduces a 1-dimensional pseudo differential operator with branching singularities by means of some functional calculus and unitary Fourier integral operators to a canonical model.