Showing papers on "Multiplication operator published in 1972"

On the comparability of ^{1/2} and ^{∗1/2}

Abstract: There exists a regularly accretive operator A in a Hubert space H such that A1'2 and A*112 have different domains. Consequently, the domain of the closed bilinear form corresponding to A is different from the domain of A112.

Invariant subspaces of the shift operator in weighted hilbert space

Abstract: A complete description is given of the closed ideals of the algebra of functions which are regular in the circle and such that , with the norm and the usual multiplication. This is equivalent to a description of the invariant subspaces of the one-sided shift operator on the weighted Hilbert space of sequences with weights (). It is shown that each closed ideal of the algebra has the form , where is the closure of in the space of functions which are regular in and continuous in with the uniform norm. Thus the ideals of the algebra have a structure similar to the structure of the ideals of the algebra : each ideal is uniquely determined by an interior function , which is the greatest common divisor of the interior parts of the functions , and the set of the common zeros of the functions .Bibliography: 19 items.

Essential spectrum for a Hilbert space operator

Abstract: Various notions of essential spectrum have been defined for densely defined closed operators on a Banach space. This paper shows that the theory for those notions of essential spectrum simplifies if the underlying space is a Hilbert space and the operator is reduced by its finite-dimensional eigenspaces. In that situation this paper classifies each essential spectrum in terms of the usual language for the spectrum of a Hilbert space operator. As an application this paper deduces the main results of several recent papers dealing with generalizations of the Weyl theorem.

Continuum eigenmodes of an inhomogeneous plasma

Abstract: The van Kampen-Case treatment of the linearized Vlasov equation is generalized to cases of non-uniform plasmas in non-uniform equilibrium fields. This is done by transforming the Vlasov operator into the representation of the eigenfunctions of the streaming operator. The Vlasov operator is thus reduced to the form of a multiplication operator perturbed by an integral operator so that the continuum eigenfunctions may be constructed by adapting a method originally devised in neutron transport theory. The calculation leads to solving a non-singular Fredholm-type integral equation. Various kinds of degeneracy of the continuous spectrum are discussed and an example is given.

The best approximation of the differentiation operator in the metric of Lp

Abstract: For Stechkin's problem of the best approximation for the differentiation operator we indicate the necessary and sufficient conditions that En be finite. We study some properties of continuous linear operators V from Lp into Lq.

Spectral function of Marchenko type for a difference operator of an even order

Abstract: The Parseval-Marchenko equality is constructed for a nonselfadjoint finite-difference operator of an arbitrary even order and it is shown how one can obtain from this, by using the matrix moment problem, the Parseval equality in the selfadjoint case.

A necessary condition for quasitriangularity

Abstract: In this note we prove that if T is a quasitriangular operator then \(T+K)=T1(T+K) for all compact operators K. A bounded linear operator £ on a Hubert space Jf is called quasitriangular if lim infp^ ||£££— ££||=0 where á2 is the directed set of all finite rank projections in S£(3tiP) under the usual ordering [3]. A(£), no(£), and iï(£) will denote the spectrum, point spectrum and approximate point spectrum of £ respectively. Theorem. // £ is quasitriangular then A(T+K) = U(T+K) for all compact operators K. Proof. Since T quasitriangular implies T+K is quasitriangular [3], we need only prove that A(£) = I1(£). Suppose that A(T)^U(T) and 2eA(£) (£). Since A(£)=n(£)un0(£*)* [2, Problem 58], A*eU0(T*). Hence £— A is bounded below and its adjoint has nontrivial null space. Thus T— A satisfies Lemma 2.1 in [1], and so T— A and hence £ is not quasitriangular, proving the Theorem. Remarks. 1. It is natural to ask whether the converse to the Theorem is also true, since A(T+K0)í¿U(T+K0) for some compact K0 is the only known criterion for proving nonquasitriangularity and since {£: A(£+A) = n(£+A) all compact K} is also uniformly closed. 2. It is easy to construct for any two compact subsets £ and M of the plane satisfying 3A/Ç Ls M an (in fact, subnormal) operator S satisfying A(S)=M, and U(S)=L. In this way, one can construct nonquasitriangular operators for which one cannot decide about the square. Received by the editors July 1, 1971. AMS 1969 subject classifications. Primary 4710, 4745.

On the Expansion of AN Entire Function of Finite Order in the Eigenfunctions of a Differential Operator

Abstract: In this paper we consider the operator which is induced in the space of entire functions of order by the operator and the boundary conditions , . Here are polynomials and is a linear functional in . We establish the completeness of the eigenfunctions of the operator , show the possibility of expansion in terms of these eigenfunctions, and estimate the rate of convergence of such an expansion.Bibliography:10 items.

Synthesis of active scattering operator by its m^{-1} -derived passive operators

Abstract: This correspondence is concerned with the synthesis of a given active scattering operator by means of a passive operator called the m^{-1} -derived passive operator (of the active operator). It is shown that by means of the m^{-1} -derived operator, an active operator can be synthesized by the well-known cascade load procedure. Furthermore, it will be shown that, like a passive operator, one can define the well-known characteristic operator function for an active operator. Relationships between the characteristic operator function of an active operator and that of its m^{-1} -derived operator are also studied.

Some Applications of Operator Valued Analytic Functions of Two Complex Variables

Abstract: This survey is meant to illustrate the main ideas of the theory which arises from a certain association of analytic operator valued functions of two complex variables wich pairs of operators related by a commutator identity.

Derivation of a Master Equation for Intramolecular Relaxation Using Projection Operator Techniques in Liouville Space

Abstract: Abstract A generalized master equation is derived to describe intramolecular rearrangement processes. It is an inhomogeneous equation, including memory effects. The derivation is based on the Liouville space formalism. Because chemically relevant information is contained in the off-diagonal elements of the density matrix, a non-diagonal coarse-graining projector is used. All necessary assumptions are stated explicitly. By making further approximations, the master equation can be reduced to an inhomogeneous von Neumann equation with an effective Liouville operator the imaginary part of which is responsible for relaxation-like coarse-grained solutions. All neglected terms are given in closed form. The character of the solutions of the master equation is discussed in \"coordinate-free\" manner, i.e. without referring to the underlying Hilbert space.

On the spectrum of a linear operator

Abstract: In the definition of the spectrum of a linear operator, it is customary to assume that the underlying space is complete. However there are occasions for which it is neither desirable nor necessary to assume completeness in order to obtain a spectral theory for an operator; for example, completeness is not needed in the Riesz theory of a compact operator (see e.g. [1: XI. 3]). Several non-equivalent definitions for the spectrum of an operator on normed spaces have appeared in the literature. We shall discuss the relationship among these definitions and some of the difficulties that arise in applying these definitions to obtain a spectral theory.

Selection of relaxation parameter

Abstract: The rate of convergence of an iteration process in Hilbert space is estimated for the purpose of solving an equation with a non-self-adjoint operator. On the assumption that the real component of the operator is positive definite, we outline the selection of the relaxation parameter and present an estimate of the rate of convergence that is exact in the class of normal operators.