Abstract: Introduction * Notation * I. On Fredholmness of Singular and Convolution Operators * II. On Fredholmness of Other Singular-Type Operators * III. Functional and Singular Integral Equations with Carleman Shifts in the Case of Continuous Coefficients * IV. Two Term Equations $(A+Qb)\Varphi = f$ with an Involutive Operator $Q$: An Abstract Approach and Applications * V. Equations with Several Generalized Involutive Operators: A Matrix Abstract Approach and Applications * VI. Application of the Abstract Approach to Singular Equations on the Real Line with Fractional-Linear Shift * Bibliography * List of Symbols * Index

Abstract: I. BASIC THEORY Chapter 1: Electromagnetic Fundamentals / Chapter 2: Introductory Functional Analysis / Chapter 3: Introductory Linear Operator Theory / Chapter 4: Spectral Theory of Linear Operators / Chapter 5: Sturm-Liouville Operators II. APPLICATIONS IN ELECTROMAGNETICS Chapter 6: Poisson's and Laplace's Boundary-Value Problems: Potential Theory / Chapter 7: Transmission Line Analysis / Chapter 8: Planarly-Layered Media Problems / Chapter 9: Cylindrical Waveguide Problems / Chapter 10: Electromagnetic Cavities

Abstract: We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model operators for both situations, linear fractional transformations for Herglotz operators, results on Friedrichs and Krein extensions, and realization theorems for classes of Herglotz operators. Moreover, we study the concrete case of Schrodinger operators on a half-line and provide two illustrations of Livsic’s result [44] on quasi-hermitian extensions in the special case of densely defined symmetric operators with deficiency indices (1,1).

Abstract: This paper characterizes boundedness, compactness and Besov cut of a composition operator from one Bloch-type space to another such space, answering a question of Madigan.

Abstract: Let X be a Banach space and let T be a bounded linear operator acting on X. Atkinson's well known theorem says that T is a Fredholm operator if and only if its projection in the algebra L(X)=F0(X) is invertible, where F0(X) is the ideal of nite rank operators in the algebra L(X) of bounded linear operators acting on X. In the main result of this paper we establish an Atkinson-type theorem for B-Fredholm operators. More precisely we prove that T is a B-Fredholm operator if and only if its projection in the algebra L(X)=F0(X) is Drazin invertible. We also show that the set of Drazin invertible elements in an algebra A with a unit is a regularity in the sense dened by Kordula and M (8).

Abstract: In this paper we prove perturbation theorems for R-sectorial operators Via the characterization of maximal Lp-regularity in terms of R-boundedness due to the second author we obtain perturbation theorems for maximal Lp-regularity in UMD -spaces We prove that R-sectoriality of A is preserved by A-small perturbations and by perturbations that are bounded in a fractional scale and small in a certain sense Here, our method seems to give new results even for sectorial operators We apply our results to uniformly elliptic systems with bounded uniformly continuous coefficients, to Schr6dinger operators with bad potentials, to the perturbation of boundary conditions, and to pseudo-differential operators with non-smooth symbols

Abstract: We consider the calculus Ψ*,* de(X, deΩ½) of double-edge pseudodifferential operators naturally associated to a compact manifold X whose boundary is the total space of a fibration. This fits into the setting of boundary fibration structures, and we discuss the corresponding geometric objects. We construct a scale of weighted double-edge Sobolev spaces on which double-edge pseudodifferential operators act as bounded operators, characterize the Fredholm elements in Ψ*,* de(X) by means of the invertibility of an appropriate symbol map, and describe a K-theoretical formula for the Fredholm index extending the Atiyah–Singer formula for closed manifolds. The algebra of operators of order (0, 0) is shown to be a Ψ*-algebra, hence its K-theory coincides with that of its C *-closure, and we give a description of the corresponding cyclic 6-term exact sequence. We define a Wodzicki-type residue trace on an ideal in Ψ*,* de(X, deΩ½), and we show that it coincides with Dixmier's trace for operators of order –dim X in ...

Abstract: Let Λ be a subset of the complex plane (in the applications this will, as a rule, be an angle with the vertex at the origin). In spectral theory it is useful to consider operators depending on a parameter λ ∈ Λ (an example of such an operator is the resolvent (A − λI)−1).

Abstract: We study the problem of approximation and representation for a family of strongly continuous operators defined in a Banach space. It allows us to extend, and in some cases to improve results from the theory ofC0-semigroups of operators to, among others, the theories of cosine families, n-times integrated semigroups, resolvent families and k-generalized solutions by means of an unified method.

Abstract: In a previous paper we have developed an axiomatic characterization of approximation operators defined by the classical diamond and box operator of modal logic. The paper presented contains the analogous results of approximation operators which are defined by using the concepts of fuzzy rough sets.

Abstract: In this paper, we show that Weyl’s theorem holds for class A operators under a certain condition. We also show that a class A operator whose Weyl spectrum equals to the one-point set {0} is always compact and normal. Mathematics subject classification (2000): 47A53, 47B20.

Abstract: Several finite difference approximations of the Dirac operator are studied and compared. Main goals are finite difference Dirac operators which allow a factorization of the discrete Laplacian. We describe the fundamental solutions of the difference operators and prove convergence results inlp-spaces. Discrete versions of the Teodorescu transform are defined.

Abstract: We study (set-valued) mappings of bounded Φ-variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the Φ-variation of a metric space valued mapping. We show that the linear span GVΦ(I;X) of the set of all mappings of bounded Φ-variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X → Y is a given mapping and the composition operator ℋ is defined by (ℋf)(t)=h(t,f(t)), where t∈I and f:I → X, we show that ℋ:GVΦ(I;X)→ GVΨ(I;Y) is Lipschitzian if and only if h(t,x)=h0(t)+h1(t)x, t∈I, x∈X. This result is further extended to multivalued composition operators ℋ with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded Φ-variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded Φ-variation.

Abstract: Parameter identification in linear ordinary differential equations examples of inverse problems exponential of a square matrix identification problems in Hilbert spaces an identification problem related to a first-order differential equation analysis of the dependence on the data of the solution to problem the minimization method a further identification problem a generalization to the vector case an identification problem for a second-order differential equation proper Riemann integrals for Banach-valued functions curvilinear integrals and Banach-valued holomorphic functions proper Riemann integrals - basic properties curvilinear integrals and Banach-valued holomorphic functions Riemann-Stieltjes integrals for Banach space-valued functions Rieman-Stieltjes integrals over compact intervals improper Riemann integrals for Banach space-valued functions improper integrals Banach algebras and spectral analysis for linear bounded operators basic properties of the spectrum of an element integration of Banach algebra-valued functions Banach algebra-valued holomorphic functions and the spectral theorem identifying parameters in first-order partial differential equations an identification problem relative to a first-order linear partial differential equation an identification problem relative to a non-linear first-order partial differential equation identification problems relative to linear bounded operators an identification problem relative to a first-order differential equation the singular case the supersingular case Continuous dependence on the data an identification problem relative to a second-order differential equation a particular case an integro-differential identification problem a one-dimensional integro-differential problem identification problems relative to linear bounded operators an abstract control problem a concrete example analysis of the continuous dependence on the data construction of an abstract model for the analysis of the continuous dependence on the data continuous dependence on the data of the solution to the identification problem Gronwall's generalized inequality linear closed operators and analytic semigroups of linear bounded operators linear closed operators resolvent set and spectrum of a linear operator sectorial operators Cauchy problems for linear abstract differential equations relative to sectorial operators and applications abstract differential equations and analytic semigroups application to Cauchy problems. (Part contents)

Abstract: HILBERT SPACES Inner Product Spaces and Hilbert Spaces Jordan-Neuman Theorem Orthogonal Decomposition of Hilbert Space Gram-Schmidt Orthonormal Procedure and its Applications FUNDAMENTAL PROPERTIES OF BOUNDED LINEAR OPERATIONS Bounded Linear Operations on Hilbert Space Partial Isometry Operator and Polar Decomposition of an Operator Polar Decomposition of an Operator and its Applications Spectrum of an Operator Numerical Range of an Operator Relations Among Several Classes of Non-normal Operators Characterizations of Convexoid Operators and Related Examples FURTHER DEVELOPMENT OF BOUNDED LINEAR OPERATORS Young Inequality and Holder-McCarthy Inequality Lowner-Heinz Inequality and Furuta Inequality Chaotic Order and the Relative Operator Entropy Aluthge Transformation on P-Hyponormal Operators and Log-Hyponormal Operators A Subclass of Paranormal Operators Including Loh-Hyponormal Operators and Several Related Classes Operator Inequalities Associated With Kantorovich Inequality and Holder-McCarthy Inequality Some Properties on Partial Isometry, Quasinormality and Paranormality Weighted Mixed Schwarz Inequality and Generalized Schwarz Inequality Selberg Inequality An Extension of Heinz-Kato Inequality Norm Inequalities Equivalent to Lower-Heinz Inequality Norm Inequalities Equivalent to Heinz Inequality Bibliography Index

Abstract: Dedication. Preface. Part 1: The Papers. On the Uncertainty Principle in Harmonic Analysis V.P. Havin. Operator Theory and Harmonic Analysis H.S. Shapiro. Probabilities and Baire's theory in harmonic analysis J.-P. Kahane. Representations of Gabor frame operators A.J.E.M. Janssen. Does Order Matter T.W. Korner. Wavelet expansions, function spaces and multifractial analysis S. Jaffard. Some Plots of Bessel Functions of Two Variables F.A. Grunbaum. Lesser Known FFT Algorithms R. Tolimieri, M. An. The Phase Problem of X-ray Crystallography H.A. Hauptman. Multiwindow Gabor-type Representations and Signal Representation by Partial Information Y.Y. Zeevi. Some polynomial extremal problems which emerged in the twentieth century B. Saffari. The Problem of Efficient Inversions and Bezout Equations N. Nikolski. Harmonic Analysis as found in Analytic Number Theory H.L. Montgomery. Mathematics of Radar B. Moran. The Mathematical Theory of Wavelets G. Weiss, E.N. Wilson. Part 2: Problems. Assorted Problems Various authors. How to Use the Fourier Transform in Asymptotic Analysis V. Gurarii, et al. Index.

Abstract: In recent papers the authors presented their approach to Feynman’s operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.

Abstract: We consider a singular differential-difference operator Λ on the real line which generalizes the Dunkl operator associated with the reflection group Z2 on R. We construct transmutation operators between Λ and the first derivative operator d/dx. We exploit these transmutation operators, firstly to determine the elementary solution of certain classes of singular differential-difference operators on a product of Euclidean spaces, and secondly to introduce a generalized translation on the real line corresponding to the operator Λ

Abstract: The resolvent (A – λ)−1 of an elliptic b-pseudodifferential operator on a compact manifold with corners (of arbitrary codimension) is shown to lie in a calculus of operators, tempered in the parameter λ in a special way. We show that the Laplace and Mellin transforms, with respect to λ, of these tempered operators can be defined and that they have Schwartz kernels which can be described geometrically. As a corollary, we obtain the structures of the kernels of the heat operator and complex powers of b-pseudodifferential operators, as the heat operator and complex powers are the Laplace and Mellin transforms, respectively, of the resolvent. The heat operator and complex powers are then used to generalize the index formula of Atiyah, Patodi, and Singer for Dirac operators on manifolds with boundary to Fredholm b-pseudodifferential operators on arbitrary compact manifolds with corners.

Abstract: For differential operators which are invariant under the action of an Abelian group Bloch theory is the preferred tool to analyze spectral properties. By shedding some new noncommutative light on this we motivate the introduction of a noncommutative Bloch theory for elliptic operators on Hilbert C*-modules. It relates properties of C*-algebras to spectral properties of module operators such as band structure, weak genericity of cantor spectra, and absence of discrete spectrum. It applies, e.g., to differential operators invariant under a projective group action, such as Schrodinger, Dirac, and Pauli operators with periodic magnetic field, as well as to discrete models, such as the almost Matthieu equation and the quantum pendulum.

Abstract: On a separable infinite dimensional complex Hilbert space, we show that the set of hypercyclic operators is dense in the strong operator topology, and moreover the linear span of hypercyclic operators is dense in the operator norm topology. Both results continue to hold if we restrict to only those hypercyclic operators with an infinite dimensional closed hyper- cyclic subspace. Our works make connections with the classical result on the nondenseness of cyclic operators in the operator norm topology, as well as the recent developments on hypercyclic subspaces.