Showing papers on "Multiplication operator published in 1990"

Nonlinear Superposition Operators

Abstract: This book is a self-contained account of knowledge of the theory of nonlinear superposition operators: a generalization of the notion of functions. The theory developed here is applicable to operators in a wide variety of function spaces, and it is here that the modern theory diverges from classical nonlinear analysis. The purpose of this book is to collect the basic facts about the superposition operator, to present the main ideas which are useful in studying its properties and to provide a comparison of its behaviour in different function spaces. Some applications are also considered, for example to control theory and optimization. Much of the work here has only appeared before in research literature, which itself is catalogued in detail here.

Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0<<\frac12

Abstract: Let A (the elastic operator) be a positive, self-adjoint operator with domain D(A) in the Hilbert space X, and let B (the dissipation operator) be another positive, self-adjoint operator satisfying ρ 1 A α 1/2α, hence differentiable on E for all t>0

A new partitioning technique for the calculation of Bloch's wave operators

Abstract: A new algorithm, involving the Bloch wave operator theory, is applied to the eigenvalue problem in large vectorial spaces. Three subspaces constituting the whole Hilbert space are then defined: the space of the trial eigenvector, an intermediate space comprising the states strongly coupled to the latter and a complementary space of large size. In the two former defined spaces each of the components of the Bloch wave operator associated with the eigenvector is separately calculated by using both procedures of standard diagonalization and of iterative perturbation. The relationship between the projections of the wave operator onto these two subspaces is deduced from a recursive scheme. The resonance state of the Henon-Heiles potential with E and A symmetry illustrate the adequacy of the formulation.

Approximate solvability of semilinear operator equations

Abstract: Sufficient conditions are given for approximate solvability of semilinear operator equations on Hilbert spaces. As a particular case, approximate controllability of a semilinear control system is dealt with.

On the spectral theory of dissipative difference operators of second order

Abstract: The boundary conditions at infinity are used in a description of all maximal dissipative extensions of the minimal symmetric operator generated in the Hilbert space by the second-order difference expression in the Weyl limit-circle case, where runs through the integer points on the half-line or the whole line, and the coefficients and are real. The characteristic functions of the dissipative extensions are computed. Completeness theorems are obtained for the system of eigenvectors and associated vectors.Bibliography: 13 titles.

A RKHS of Entire Functions and its Multiplication Operator. An Explicit Example

Abstract: The theory of unbounded subnormal operators differs in many aspects from its bounded counterpart (see [2], [3] and [4] for systematic studies of the subject). In this report we wish to make transparent one of these aspects giving an explicit example of a subnormal operator in a Reproducing Kernel Hilbert Space of entire functions, which comes from some indeterminate Stieltjes moment problem [1]. This leads to an analytic 2 subnormal operator having normal extensions in two, of quite different nature, L 2 spaces; one of the spaces involves a measure absolutely continuous with respect to the planar Lebesgue one, while the other is concentrated on a countable number of circles around the origin, whose radii go both to zero and to infinity. In other words, relating them to the bounded case, the first of these spaces looks like the Bergman one, the other reminds the Hardy space.

Optimization of adaptive direct methods for the solution of operator equations in Hilbert space

Abstract: Some general theorems on estimates of error of optimal adaptive direct methods of solution of operator equations of type II in Hilbert space are obtained.

Theory of operator means

Abstract: New properties of operator connections and means are established. Specifically, representations of an arbitrary connection by means of a concave representing function, an estimate of the norm of a connection in a von Neumann-Schatten ideal, a relation between operator means and convolutions onto operator domains are obtained.

Spectral pictures of hyponormal bilateral operator weighted shifts

Abstract: This note provides a complete description of the spectral picture of a hyponormal bilateral operator weighted shift of finite multiplicity. If the weights are m x m matrices, then the operator cannot by cyclic, unless it is a normal operator of a very special kind. An example shows that, nevertheless, there exists a cyclic, nonnormal, hyponormal bilateral operator weighted shift whose weights are operators acting on an infinite dimensional Hubert space. (This answers a question of J. B. Conway.)

The minimal normal extension for _{} on the Hardy space of a planar region

Abstract: Multiplication by the independent variable on H2(R) for R a bounded open region in the complex plane C is a subnormal operator. This paper characterizes its minimal normal extension N. Any normal operator is determined by a scalar-valued spectral measure and a multiplicity function. It is a consequence of some standard operator theory that a scalar-valued spectral measure for N is harmonic measure for R, co. This paper investigates the multiplicity function m for N. It is shown that m is bounded above by two wo-a.e., and necessary and sufficient conditions are given for m to attain this upper bound on a set of positive harmonic measure. Examples are given which indicate the relationship between N and the boundary of R.

Projection operator in the boson expansion techniques.

Abstract: The projection operator onto the physical space is investigated in the boson expansion techniques. An explicit expression for the projection operator contains many body terms. From this expression, recurrent relations are obtained. Based on the properties of the projection operator, a new general form of boson mapping is derived.

A simple iterative method for solving linear operator equations

Abstract: The necessary and sufficient conditions are obtained for the convergence of the simple iterative method that apply in the domain of location of the spectrum of an operator in the complex plane, and the possibility of an algorithmic definition of the iteration parameters when calculating the iterations is considered.

A Compact Exponential Operator for Generating Rotating and Squeezing Transformations

Abstract: We point out that the compact exponential operator (where are real; and are momentum and coordinate operators, respectively; the units of is taken, and are harmonic oscillator's mass and frequency, respectively) is a generalized single-mode squeeze operator which can generate mixed squeezing and rotating transformations. The decomposition of as a product of a squeeze operator and a rotating operator is derived through the technique of integration within an ordered product. The fluctuation in quadrature phases for the squeezed vacuum state generated by is analyzed.

Block‐diagonalization in second quantization

Abstract: The unitary matrix that brings a Hermitian H into block‐diagonal form can be uniquely determined under very simple and transparent conditions. Int his work the block‐diagonalization problem is investigated in the framework of the second quantization formalism. Starting with an operator H which in any n‐particle Fock space has a well‐defined matrix representation an attemt was made to answer the question whether the transformation matrices T which can be separately given in the various n‐particle spaces can be considered matrix representations of the same operator. T. Interestingly, the very important result was reached that the block‐diagonalization operator T exists and is unique. As a particular example, attention was concentrated on the case of an operator H given by a one‐particle operator. In this case the block‐diagonalization operator can be constructed and given in explicit from. This approach is applied to the theory of Green’s functions where the block‐diagonalization of the Hamiltonian has interesting consequences that are illustrated in some details.