Abstract: Jacobi operators: Jacobi operators Foundations of spectral theory for Jacobi operators Qualitative theory of spectra Oscillation theory Random Jacobi operators Trace formulas Jacobi operators with periodic coefficients Reflectionless Jacobi operators Quasi-periodic Jacobi operators and Riemann theta functions Scattering theory Spectral deformations-Commutation methods Completely integrable nonlinear lattices: The Toda system The initial value problem for the Toda system The Kac-van Moerbeke system Notes on literature Compact Riemann surfaces-A review Hergoltz functions Jacobi difference equations with MathematicaR Bibliography Glossary of notations Index.

Abstract: Introduction to C*-algebras Normal operators Compact operators Some non-normal operators More on C*-algebras Compact perturbations Introduction to von Neumann algebras Reflexivity Bibliography Index List of symbols.

Abstract: In this paper we fill some gaps in the arguments of our previous papers [1,2] In particular, we give a proof that the L operators of Conformal Field Theory indeed satisfy the defining relations of the Yang–Baxter algebra Among other results we present a derivation of the functional relations satisfied by T and Q operators and a proof of the basic analyticity assumptions for these operators used in [1,2]

Abstract: For a finite reflection group on $\b R^N,$ the associated Dunkl operators are parametrized first-order differential-difference operators which generalize the usual partial derivatives They generate a commutative algebra which is - under weak assumptions - intertwined with the algebra of partial differential operators by a unique linear and homogeneous isomorphism on polynomials In this paper it is shown that for non-negative parameter values, this intertwining operator is positivity-preserving on polynomials and allows a positive integral representation on certain algebras of analytic functions This result in particular implies that the generalized exponential kernel of the Dunkl transform is positive-definite

Abstract: We study a class of bounded linear operators acting on a Banach spaceX called B-Fredholm operators. Among other things we characterize a B-Fredholm operator as the direct sum of a nilpotent operator and a Fredholm operator and we prove a spectral mapping theorem for B-Fredholm operators.

Abstract: In this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting Lax representation. Several new examples are given. In particular, we find the recursion operators for some KdV-type systems of integrable equations.

Abstract: We give a sufficient condition for singular integral operators and, more generally, Calderon-Zygmund operators to satisfy the weak (p, p) inequality u({x ∈ R : |Tf(x)| > t}) ≤ C tp ∫ R n |f |pv dx, 1 0. This conditions is stronger than the Ap condition and is sharp since it fails when δ = 0.

Abstract: In this paper we apply the theory of semigroups of operators in order to obtain the existence and uniqueness of solutions for the mixed initial-boundary value problems in thermoelasticity of dipolar bodies. The continuous dependence of the solutions upon initial data and supply terms is also proved.

Abstract: In this paper, we study boundedness of integral operators on gen- eralized Morrey spaces and its application to estimates in Morrey spaces for the Schrodinger operator L2 = -A + V(x) + W(x) with nonnegative V E (RH).,

Abstract: Bilocal light-ray operators which are Lorentz scalars, vectors or antisymmetric tensors, and which appear in various hard scattering QCD processes, are decomposed into operators of definite twist. These operators are harmonic tensor functions and their Taylor expansion consists of (traceless) local light-cone operators with span irreducible representations of the Lorentz group with definite spin j and common geometric twist (= dimension - spin). Some applications concerning the nonforward matrix elements of these operators and the generalization fo conformal light-cone operators of definite twist is considered. The group theoretical background of the method has been made clear.

Abstract: A bounded linear operator A on a Hilbert space H with inner product (·, ·) is said to be hyponormal if its selfcommutator [A∗, A] = A∗A − AA∗ induces a positive semidefinite quadratic form on H via ξ 7→ ([A∗, A]ξ, ξ), for ξ ∈ H. Let H(T) denote the Hardy space of the unit circle T = ∂D in the complex plane. Recall that given φ ∈ L∞(T), the Toeplitz operator with symbol φ is the operator Tφ on H(T) defined by Tφf = P (φ · f), where f ∈ H(T) and P denotes the projection that maps L(T) onto H(T). The hyponormality of Toeplitz operators has been studied by C. Cowen [1],[2], P. Fan [4], C. Gu [8], T. Ito and T. Wong [9], T. Nakazi and K. Takahashi [11], D. Yu [13], K. Zhu [14], R. Curto, D. Farenick, the second and the third named authors [3],[5],[6],[10] and others. An elegant theorem of C. Cowen [2] characterizes the hyponormality of a Toeplitz operator Tφ on H(T) by properties of the symbol φ ∈ L∞(T). K. Zhu [14] reformulated Cowen’s criterion and then showed that the hyponormality of Tφ with polynomial symbols φ can be decided by a method based on the classical interpolation theorem of I. Schur [12]. Also Farenick and the third named author [5] characterized the hyponormality of Tφ in terms of the Fourier coefficients of the trigonometric polynomial φ in the cases that the outer coefficients of φ have the same modulus. But the case of arbitrary trigonometric polynomials φ, though solved in principle by Cowen’s theorem or Zhu’s theorem, is in practice very complicated. On the other hand, Nakazi and Takahashi [11, Corollary 5] showed that if φ(z) = ∑N n=−m anz n is a trigonometric polynomial with m ≤ N and if for every zero ζ of zφ such that |ζ| > 1, the number 1/ζ is a zero of zφ in the open unit disk D of multiplicity greater than or equal to the multiplicity of ζ, then Tφ is hyponormal. But the converse is not true in general. To see this consider the following trigonometric polynomial: φ(z) = z−2(z−2)(z−1)(z− 15 )(z− 13 ). Then φ(z) = 2 15z −2 − 19 15z + 55 15 − 53 15z + z. Using an argument of P. Fan [4, Theorem 1] – for every trigonometric polynomial φ of the form φ(z) = ∑2 n=−2 anz ,

Abstract: We construct parametrized Yang-Baxter operators from algebra struc-tures, transfer the theory to coalgebras, and find the cases where such operators arising from (co) algebras are isomorphic. We give examples in dimension 2.

Abstract: We show that for a generic deformation of a linear analytic differential equation with an irregular singularity of order 1 of a generic (nonresonant) type, Stokes operators of the nonperturbed equation are limits of transition operators between appropriate eigenbases of the monodromy operators of the perturbed equation. We prove a generalization of this statement for arbitrary degree nonresonant irregular singularity.

Abstract: Abstract Recent work of Ding and Huang shows that if we perturb a bounded operator (between Hilbert spaces) which has closed range, then the perturbed operator again has closed range. We extend this result by introducing a weaker perturbation condition, and our result is then used to prove a theorem about the stability of frames for a subspace.

Abstract: C*-algebras: C*-algebras: Definitions and examples C*-algebras: Constructions Positivity in C*-algebras K-theory I Tensor products of C*-algebras Crossed products I Crossed products II: Examples Free products K-theory II: Roots in topology and index theory C*-algebraic K-theory made concrete, or trick or treat with $2 \times 2$ matrix algebras Dilation theory C*-algebras and mathematical physics C*-algebras and several complex variables von Neumann algebras: Basic structure of von Neumann algebras von Neumann algebras (Type $II_1$ factors) The equivalence between injectivity and hyperfiniteness, part I The equivalence between injectivity and hyperfiniteness, part II On the Jones index Introductory topics on subfactors The Tomita-Takesaki theory explained Free products of von Neumann algebras Semigroups of endomorphisms of $\mathcal{B}(H)$ Classification of C*-algebras AF-algebras and Bratteli diagrams Classification of amenable C*-algebras I Classification of amenable C*-algebras II Simple AI-algebras and the range of the invariant Classification of simple purely infinite C*-algebras I Hereditary subalgebras of certain simple non real rank zero C*-algebras: Preface Introduction The isomorphism theorem The range of the invariant Bibliography Paths on Coxeter diagrams: From platonic solids and singularities to minimal models and subfactors: Preface/Acknowledgements The Kauffman-Lins recoupling theory Graphs and connections An extension of the recoupling model Relations to minimal models and subfactors Bibliography.

Abstract: A system of ordinary differential operators of mixed order on an interval (0,τ0), ro0 > 0, is considered, where the coefficients may be singular at 0. A special case has been dealt with by Kako where the essential spectrum of an operator associated with a linearized magnetohydrodynamic equation was explicitly calculated. In the present first part of the paper we study an almost regular special case which can be treated by the operator theoretical methods developed by Atkinson, Langer, Mennicken and Shkalikov. A closed linear operator is associated with the given system of differential operators and its essential spectrum is explicitly characterized in terms of the coefficients of these differential operators.

Abstract: of the real line. A necessary and sufficient condition is given for these operators to be bounded, and a characterisation is given for the operatorbounds. There are applications of the results to integral inequalities; also to properties of the domains of self-adjoint unbounded operators, in Hilbert function spaces, associated with the classical orthogonal polynomials and their generalisations.