Abstract: Linear operations in Banach spaces Entropy numbers, s-numbers, and eigenvalues Unbounded linear operators Sesquilinear forms in Hilbert spaces Sobolev spaces Generalized Dirichlet and Neumann boundary-value problems Second-order differential operators on arbitrary open sets Capacity and compactness criteria Essential spectra Essential spectra of general second-order differential operators Global and asymptotic estimates for the eigen-values of - + q when q is real. Estimates for the singular values of - + q when 1 is complete Bibliography Notation index Subject index

Abstract: Introduction Analysis Background A Menagerie of Spaces Some Theorems on Integration Geometric Function Theory in the Disk Iteration of Functions in the Disk The Automorphisms of the Ball Julia-Caratheodory Theory in the Ball Norms Boundedness in Classical Spaces on the Disk Compactness and Essential Norms in Classical Spaces on the Disk Hilbert-Schmidt Operators Composition Operators with Closed Range Boundedness on Hp (BN) Small Spaces Compactness on Small Spaces Boundedness on Small Spaces Large Spaces Boundedness on Large Spaces Compactness on Large Spaces Hilbert-Schmidt Operators Special Results for Several Variables Compactness Revisited Wogen's Theorem Spectral Properties Introduction Invertible Operators on the Classical Spaces on the Disk Invertible Operators on the Classical Spaces on the Ball Spectra of Compact Composition Operators Spectra: Boundary Fixed Point, j'(a)

Abstract: -Preface for the Instructor-Preface for the Student-Acknowledgments-1. Vector Spaces- 2. Finite-Dimensional Vector Spaces- 3. Linear Maps- 4. Polynomials- 5. Eigenvalues, Eigenvectors, and Invariant Subspaces- 6. Inner Product Spaces- 7. Operators on Inner Product Spaces- 8. Operators on Complex Vector Spaces- 9. Operators on Real Vector Spaces- 10. Trace and Determinant-Photo Credits-Symbol Index-Index.

Abstract: This correspondence deals with the notion of connected operators. Starting from the definition for operator acting on sets, it is shown how to extend it to operators acting on function. Typically, a connected operator acting on a function is a transformation that enlarges the partition of the space created by the flat zones of the functions. It is shown that from any connected operator acting on sets, one can construct a connected operator for functions (however, it is not the unique way of generating connected operators for functions). Moreover, the concept of pyramid is introduced in a formal way. It is shown that, if a pyramid is based on connected operators, the flat zones of the functions increase with the level of the pyramid. In other words, the flat zones are nested. Filters by reconstruction are defined and their main properties are presented. Finally, some examples of application of connected operators and use of flat zones are described. >

Abstract: Pseudodifferential operators arise naturally in the solution of boundary problems for partial differential equations. The formalism of these operators serves to make the Fourier-Laplace method applicable for nonconstant coefficient equations. This book presents the technique of pseudodifferential operators and its applications, especially to the Dirac theory of quantum mechanics. The treatment uses 'Leibniz formulas' with integral remainders or as asymptotic series. A pseudodifferential operator may also be described by invariance under action of a Lie-group. The author discusses connections to the theory of C*-algebras, invariant algebras of pseudodifferential operators under hyperbolic evolution and the relation of the hyperbolic theory to the propagation of maximal ideals. This book will be of particular interest to researchers in partial differential equations and mathematical physics.

Abstract: Matrix-valued functions functions of compact operators functions of nonself-adjoint operators perterbations of finite dimensional and compact operators perterbations of noncompact operators perterbations of operators on a tensor product of Hilbert spaces stability and boundedness of ordinary differential systems stability of retarded systems absolute stability of solutions of Voletta integral equations stability of semilinear parabolic systems stability of Volterra integrodifferential systems and applications to viscoelasticity semilinear boundary value problems list of main symbols.

Abstract: A class of operators related to Hermite-Pade approximants is defined. The spectral analysis of these operators is connected with the asymptotic behavior of polynomials defined by systems of orthogonality relations.Bibliography: 39 titles.

Abstract: We study some algebras generated by the singular integral operator with the quaternionic Cauchy kernel and the multiplication operators by continuous or piece-wise continuous functions. This allows us to describe the Fredholm theory of hyperholomorphic Riemann boundary value problems whose formulations take into account both the non-commutativity of the quaternionic multiplication and the existence of various classes of hyperholomorphic functions. Certain classes of hyperholomorphic Toenlitz operators are also studied.

Abstract: For any finite reflection group G on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections in G. There exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients, for all but a singular set of parameter values (containing only certain negative rational numbers). This paper constructs an integral transform implementing the intertwining operator for the group S3, the symmetric group on three objects, for parameter value > ! . The transform is realized as an absolutely continuous measure on a compact subset of M2(R), which contains the group as a subset of its boundary. The construction of the integral formula involves integration over the unitary group U(3) . Associated to any finite reflection group G on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections in G. It has been shown that there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients, for all but a "singular set" of parameter values. This singular set contains only negative values and is closely linked to the zero-set of the Poincare series of G. This paper constructs an integral transform implementing the intertwining map for the group S3, the symmetric group on three objects, for positive parameter values. Previously this had been done only for the group Z2 (acting by sign-change on R) where the transform is a classical fractional integral. The transform in this paper has its origin in the adjoint action of the unitary group U(3) on the linear space of real diagonal 3 x 3 matrices (the complexification of the maximal torus). This will lead to a transform realized as an absolutely continuous measure on a certain compact subset of M2(R). Here is a concise statement of the main result (rephrased from formulas (5.1), (5.6)): the operator V intertwines the differential-difference operators Received by the editors August 22, 1994. 1991 Mathematics Subject Classification. Primary 22E30, 33C80; Secondary, 33C50, 20B30.

Abstract: Mathematical preliminaries, basic definitions and theorems from the theory of boundary value problems and semigroups of operators in Banach spaces formal asymptotic expansion, singularly perturbed evolution equations of the resonance type standard asymptotic expansion general theory of singularly perturbed evolution equations with bounded operators, bounded generators applications to the kinetic theory, general properties of linear kinetic equations, existence theorems, equations with bounded operators of resonance type, equations with unbounded operators of resonance type nonlinear equations, selected examples of nonlinear kinetic equations.

Abstract: Introduction Some notions connected with group actions Some notions and results connected with elliptic operators Elliptic operators and group actions Positive multiplicative solutions Nilpotent groups: extreme points and multiplicative solutions Nonnegative solutions of parabolic equations Invariant operators on a manifold with boundary Examples and open problems Appendix: analyticity of $\Lambda (\xi, \scr L)$ References.

Abstract: We use the method of self-adjoint extensions to define a self-adjoint operator AT as the singular perturbation of a given self-adjoint operator A by a singular operator T on a Hilbert space. We also find the structure of a singular operator Q such that the singular perturbation of A2 by Q satisfies (A2)Q = (AT)2. We obtain the explicit form of Q in terms of A and T. A definition of the n-th power for strictly positive symmetric operators is also given.

Abstract: We discuss minisuperspace aspects of a nonempty Robertson-Walker universe containing a scalar matter field. The requirement that the Wheeler-DeWitt (WDW) operator be self-adjoint is a key ingredient in constructing the physical Hilbert space and has nontrivial cosmological implications since it is related to the problem of time in quantum cosmology. Namely, if time is parametrized by matter fields we find two types of domains for the self-adjoint WDW operator: a nontrivial domain is comprised of zero current (Hartle-Hawking-type) wave functions and is parametrized by two new parameters, whereas the domain of a self-adjoint WDW operator acting on tunneling (Vilenkin-type) wave functions is a single ray. On the other hand, if time is parametrized by the scale factor both wave function types give rise to nontrivial domains for the self-adjoint WDW operators, and no new parameters appear in them.

Abstract: A boundary-value problem for the non-self-adjoint differential operators with a regular singularity at zero is investigated. Theorems are obtained on completeness, on the expansion with respect to the eigen- and associated functions of the boundary-value problem on a finite interval, and on equiconvergence. In addition, the inverse problem is investigated.

Abstract: This paper reports expository talks, presented at the NATO-ASI, on scattered data interpolation by means of positive linear operators, relating to classical and extended operators of Shepard’s type. Emphasis is placed on some topics such as constructive procedures, convergence and rate of approximation, connection with physical models, computational problems, algorithms for parallel, multistage and recursive computation.

Abstract: Product and summation operators for predicate transformers were introduced by Naumann and by Martin using category theoretic considerations. In this paper, we formalise these operators in the higher order logic approach to the refinement calculus of Back and von Wright, and examine various algebraic properties of these operators. There are several motivating factors for this analysis. The product operator provides a model of simultaneous execution of statements, while the summation operator provides a simple model of late binding. We also generalise the product operator slightly to form an operator that corresponds to conjunction of specifications. We examine several applications of the these operators showing, for example, how a combination of the product and summation operators could be used to model inheritance in an object-oriented programming language.

Abstract: some regularity conditions on \\Omega are assumed (see [2]). The bounded ness of T with the rough condition \\Omega\\in L^{q}(S^{n-1}\\cross S^{m-1}) instead of regularity is obtained in [1]. In this paper, we shall use the method of block decomposition for functionsu to improve the result of boundedness above. It should be pointed out that the method of block decomposition for functions is originated by M. H. Taibleson and G. Weiss in the study of the convergence of the Fourier series (see [6]). Latter on, many applications of the block decomposition to Harmonic analysis were discovered (see [5]). For example, a sort of method related to block decompositions is applied to study the boundedness of singular integral operators with rough kernel in [3]-[4] . Thus, this paper can also be regarded as generalization of the one-parameter results in [3]-[4] . Let us begin with the definition of q -block on S^{n-1}\\cross S^{m-1} .

Abstract: We study nonsymmetric tridiagonal operators acting in the Hilbert space ?2 and describe the spectrum and the resolvent set of such operators in terms of a continued fraction related to the resolvent. In this way we establish a connection between Pade approximants and spectral properties of nonsymmetric tridiagonal operators.