Abstract: 1. Preliminaries.- 2. Partial Differential Operators with Constant Coefficients.- 3. L2 Sobolev Spaces.- 4. Basic Theory of Pseudo Differential Operators.- 5. Lp and Lipschitz Estimates.- References.

Abstract: Ellipticity condition, proper ellipticity and Lopatinskii condition imply the Fredholm property of elliptic problems in bounded domains In addition, invertibility of limiting problems determines the Fredholm property and solvability conditions of elliptic problems in unbounded domains If this property is satisfied, then the index of the operator is defined There is an extensive literature devoted to the index of elliptic operators in bounded domains and for some classes of operators in unbounded domains (see the bibliographical comments)

Abstract: We advance a correspondence between the topological defect operators in Liouville and Toda conformal field theories — which we construct — and loop operators and domain wall operators in four dimensional $ \mathcal{N} = 2 $ supersymmetric gauge theories on S 4 Our computation of the correlation functions in Liouville/Toda theory in the presence of topological defect operators, which are supported on curves on the Riemann surface, yields the exact answer for the partition function of four dimensional gauge theories in the presence of various walls and loop operators; results which we can quantitatively substantiate with an independent gauge theory analysis As an interesting outcome of this work for two dimensional conformal field theories, we prove that topological defect operators and the Verlinde loop operators are different descriptions of the same operators

Abstract: 1 Introduction.- 2 Slice monogenic functions.- 3 Functional calculus for n-tuples of operators.- 4 Quaternionic Functional Calculus.- 5 Appendix: The Riesz-Dunford functional calculus.- Bibliography.- Index.

Abstract: Received by the editors March 8, 2010 and, in revised form, August 2, 2010, October 26, 2010, and December 17, 2010. 2010 Mathematics Subject Classification. Primary 35B65, 45G05, 47G10.

Abstract: We define cut-and-join operators in Hurwitz theory for merging two branch points of an arbitrary type. These operators have two alternative descriptions: (1) the GL characters are their eigenfunctions and the symmetric group characters are their eigenvalues; (2) they can be represented as W-type differential operators (in particular, acting on the time variables in the Hurwitz-Kontsevich τ-function). The operators have the simplest form when expressed in terms of the Miwa variables. They form an important commutative associative algebra, a universal Hurwitz algebra, generalizing all group algebra centers of particular symmetric groups used to describe the universal Hurwitz numbers of particular orders. This algebra expresses arbitrary Hurwitz numbers as values of a distinguished linear form on the linear space of Young diagrams evaluated on the product of all diagrams characterizing particular ramification points of the branched covering.

Abstract: 1 Experiments, Measure, and Integration- A Measures- Experiments and weight functions, expected value of a weight function, measures, Lebesgue measure, signed measures, complex measures, measurable functions, almost everywhere equality- B Integration- Simple functions, simple integrals, general integrals, Lebesgue integrals, properties of integrals, expected values as integrals, complex integrals- 2 Hilbert Space Basics- Inner product space, norm, orthogonality, Pythagorean theorem, Bessel and Cauchy-Schwarz and triangle inequalities, Cauchy sequences, convergence in norm, completeness, Hilbert space, summability, bases, dimension- 3 The Logic of Nonclassical Physics- A Manuals of Experiments and Weights- Manuals, outcomes, events, orthogonality, refinements, compatibility, weights on manuals, electron spin, dispersion-free weights, uncertainty- B Logics and State Functions- Implication in manuals, logical equivalence, operational logic, implication and orthocomplementation in the logic, lattices, general logics (quantum logics), propositions, compatibility, states on logics, pure states, epistemic and ontological uncertainty- 4 Subspaces in Hilbert Space- Linear manifolds, closure, subspaces, spans, orthogonal complements, the subspace logic, finite projection theorem, compatibility of subspaces- 5 Maps on Hilbert Spaces- A Linear Functional and Function Spaces- Linear maps, continuity, boundedness, linear functional, Riesz representation theorem, dual spaces, adjoints, Hermitian operators- B Projection Operators and the Projection Logic- Projection operators, summability of operators, the projection logic, compatibility and commutativity- 6 State Space and Gleason's Theorem- A The Geometry of State Space- State space, convexity, faces, extreme points, properties, detectability, pure states, observables, spectrum, expected values, exposed faces- B Gleason's Theorem- Vector state, mixture, resolution of an operator into projection operators, expected values of operators, Gleason's theorem- 7 Spectrality- A Finite Dimensional Spaces, the Spectral Resolution Theorem- Eigenvalues, point spectrum, eigenspaces, diagonalization, the spectral resolution theorem- B Infinite Dimensional Spaces, the Spectral Theorem- Spectral values, spectral measures, the spectral theorem, functions of operators, commutativity and functional relationships between operators, commutativity and compatibility of operators- 8 The Hilbert Space Model for Quantum Mechanics and the EPR Dilemma- A A Brief History of Quantum Mechanics- B A Hilbert Space Model for Quantum Mechanics- Schroedinger's equation, probability measures, stationary states, the harmonic oscillator, the assumptions of quantum mechanics, position and momentum operators, compatibility- C The EPR Experiment and the Challenge of the Realists- Electron spin, spin states, singlet state, EPR apparatus, the EPR dilemma- Index of Definitions

Abstract: One of the major unsolved problems in operator theory is the fifty-year-old invariant subspace problem, which asks whether every bounded linear operator on a Hilbert space has a nontrivial closed invariant subspace. This book presents some of the major results in the area, including many that were derived within the past few years and cannot be found in other books. Beginning with a preliminary chapter containing the necessary pure mathematical background, the authors present a variety of powerful techniques, including the use of the operator-valued Poisson kernel, various forms of the functional calculus, Hardy spaces, fixed point theorems, minimal vectors, universal operators and moment sequences. The subject is presented at a level accessible to postgraduate students, as well as established researchers. It will be of particular interest to those who study linear operators and also to those who work in other areas of pure mathematics.

Abstract: In this paper we present the sequence of linear Bernstein-type operators defined for f ∈ C [ 0 , 1 ] by B n ( f ∘ τ − 1 ) ∘ τ , B n being the classical Bernstein operators and τ being any function that is continuously differentiable ∞ times on [ 0 , 1 ] , such that τ ( 0 ) = 0 , τ ( 1 ) = 1 and τ ′ ( x ) > 0 for x ∈ [ 0 , 1 ] . We investigate its shape preserving and convergence properties, as well as its asymptotic behavior and saturation. Moreover, these operators and others of King type are compared with each other and with B n . We present as an interesting byproduct sequences of positive linear operators of polynomial type with nice geometric shape preserving properties, which converge to the identity, which in a certain sense improve B n in approximating a number of increasing functions, and which, apart from the constant functions, fix suitable polynomials of a prescribed degree. The notion of convexity with respect to τ plays an important role.

Abstract: Preface.- The Finite Fourier Transform.- Translation-Invariant Linear Operators.- Circulant Matrices.- Convolution Operators.- Fourier Multipliers.- Eigenvalues and Eigenfunctions.- The Fast Fourier Transform.- Time-Frequency Analysis.- Time-Frequency Localized Bases.- Wavelet Transforms and Filter Banks.- Haar Wavelets.- Daubechies Wavelets.- The Trace.- Hilbert Spaces.- Bounded Linear Operators.- Self-Adjoint Operators.- Compact Operators.- The Spectral Theorem.- Schatten-von Neumann Classes.- Fourier Series.- Fourier Multipliers on S1.- Pseudo-Differential Operators on S1.- Pseudo-Differential Operators on Z.- Bibliography.- Index.

Abstract: We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.

Abstract: We use the method of similar operators to study the spectral properties of Dirac operators, and obtain results on the asymptotic behaviour of the spectra of Dirac operators and the convergence of spectral expansions.

Abstract: In this paper we derive the most general first-order symmetry operator commuting with the Dirac operator in all dimensions and signatures. Such an operator splits into Clifford even and Clifford odd parts which are given in terms of odd Killing-Yano and even closed conformal Killing-Yano inhomogeneous forms, respectively. We study commutators of these symmetry operators and give necessary and sufficient conditions under which they remain of the first-order. In this specific setting we can introduce a Killing-Yano bracket, a bilinear operation acting on odd Killing-Yano and even closed conformal Killing-Yano forms, and demonstrate that it is closely related to the Schouten-Nijenhuis bracket. An important nontrivial example of vanishing Killing-Yano brackets is given by Dirac symmetry operators generated from the principal conformal Killing-Yano tensor [hep-th/0612029]. We show that among these operators one can find a complete subset of mutually commuting operators. These operators underlie separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all dimensions [arXiv:0711.0078].

Abstract: We study tent spaces on general measure spaces $(\Omega, \mu)$. We assume that there exists a semigroup of positive operators on $L^p(\Omega, \mu)$ satisfying a monotone property but do not assume any geometric/metric structure on $\Omega$. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman's $H^1$-BMO duality theory. We also get a $H^1$-BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative $L^p$ spaces.

Abstract: Preface * Introduction * Spaces of test functions * Schwartz distributions * Calculus for distributions * Distributions as derivatives of functions * Tensor products * Convolution products * Applications of convolution * Holomorphic functions * Fourier Transformation * Distributions and analytic functions * Other spaces of generalized functions * Hilbert spaces: A brief historical introduction * Inner product spaces and Hilbert spaces * Geometry of Hilbert spaces * Separable Hilbert spaces * Direct sums and tensor products * Topological aspects * Linear operators * Quadratic forms * Bounded linear operators * Special classes of bounded operators * Self-adjoint Hamilton operators * Elements of spectral theory * Spectral theory of compact operators * The spectral theorem * Some applications of the spectral representation * Introduction * The direct methods in the calculus of variations * Differential calculus on Banach spaces and extrema of differentiable functions * Constrained minimization problems (Method of Lagrange multipliers) * Boundary and eigenvalue problems * Density functional theory of atoms and molecules * Appendices * References * Index

Abstract: In this paper, we introduce a generalization of the Bernstein- Schurer operators based on q-integers and get a Bohman-Korovkin type approximation theorem of these operators. We also compute the rate of convergence by using the first modulus of smoothness.

Abstract: Here we give some approximation theorems concerning pointwise convergence for nets of nonlinear integral operators of the form: where the kernel (K λ)λ∈Λ satisfies some general homogeneity assumptions. Here Λ is a nonempty set of indices provided with a topology.

Abstract: 1. Introduction. 2. Introduction to Metric Spaces. 3. Energy Spaces and Generalized Solutions. 4. Approximation in a Normed Linear Space. 5. Elements of the Theory of Linear Operators. 6. Compactness and Its Consequences. 7. Spectral Theory of Linear Operators. 8. Applications to Inverse Problems. Index.

Abstract: Let A, B, X, and Y be bounded linear operators on a complex Hilbert space. It is shown that where w(·) and ‖·‖ are the numerical radius and the usual operator norm, respectively. This inequality includes and improves upon earlier numerical radius inequalities proved in this context. Applications of this inequality are given to obtain new numerical radius inequalities for commutators of self-adjoint and positive operators.

Abstract: In this paper we prove L p -boundedness properties of spectral multipliers associated with multidimensional Bessel operators In order to do this we estimate the L p -norm of the imaginary powers of Bessel operators We also prove that the Hankel multipliers of Laplace transform type on (0,∞) n are principal value integral operators of weak type (1,1)

Abstract: This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension $d\geq 3$ the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in presence of embedded eigenvalues and threshold singularities.

Abstract: By introducing the concept of limited completely continuous op- erators between two arbitrary Banach spaces X and Y , we give some properties of this concept related to some well known classes of operators and specially, related to the Gelfand-Phillips property of the space X or Y. Then some necessary and sucient conditions for the Gelfand-Phillips property of closed subspace M of some operator spaces, with respect to limited complete conti- nuity of some operators on M, so-called, evaluation operators, are verified.