Showing papers on "Operator theory published in 1981"
TL;DR: In this article, the scattering theory of a conservative nonlinear one-parameter group of operators on a Hilbert space X relative to a group of linear unitary operators is studied.
594 citations
Book•
1 Jan 1981
TL;DR: In this article, the authors explore the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds.
Abstract: The theory of Toeplitz operators has come to resemble more and more in recent years the classical theory of pseudodifferential operators For instance, Toeplitz operators possess a symbolic calculus analogous to the usual symbolic calculus, and by symbolic means one can construct parametrices for Toeplitz operators and create new Toeplitz operators out of old ones by functional operationsIf P is a self-adjoint pseudodifferential operator on a compact manifold with an elliptic symbol that is of order greater than zero, then it has a discrete spectrum Also, it is well known that the asymptotic behavior of its eigenvalues is closely related to the behavior of the bicharacteristic flow generated by its symbolIt is natural to ask if similar results are true for Toeplitz operators In the course of answering this question, the authors explore in depth the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds
579 citations
Book•
1 Jan 1981
TL;DR: In this paper, Wigner and Racah operators are associated with W-Algebra, an algebra of invariant operators, and a list of symbols Indices is given.
Abstract: 1. Introduction 2. Algebraic structures associated with Wigner and Racah operators 3. Null space properties and structure theorems for RW-Algebra 4. W-Algebra: an algebra of invariant operators 5. Special topics Appendix List of symbols Indices.
307 citations
Book•
1 Jan 1981
TL;DR: In this article, the authors provide an introduction to functional analysis with an emphasis on the theory of linear operators and its application to differential and integral equations, approximation theory, and numerical analysis.
Abstract: This text provides an introduction to functional analysis with an emphasis on the theory of linear operators and its application to differential and integral equations, approximation theory, and numerical analysis. It begins with the geometry of Hilbert space and proceeds to the spectral theory for compact self-adjoint operators with a wide range of applications. Part of the volume is devoted to Banach spaces and operators acting on these spaces. Presented as a natural continuation of linear algebra, this book aims to provide a foundation in operator theory, an essential part of mathematical training for students of mathematics, engineering, and other technical services.
258 citations
1 Jan 1981
189 citations
73 citations
TL;DR: A survey of papers reviewed in Ref. Zh. Matematika from 1953 to 1978 on the theory of linear operators in (mainly Hibert) spaces with indefinite metric and their applications to various domains of mathematics and mechanics is presented in this article.
Abstract: The paper contains a survey of papers reviewed in Ref. Zh. Matematika from 1953–1978 on the theory of linear operators in (mainly Hibert) spaces with indefinite metric and their applications to various domains of mathematics and mechanics. As a preliminary, the needed results on the geometry of spaces with indefinite metric are described.
68 citations
TL;DR: In this article, a survey of recent results on essential selfadjointness problems for linear differential operators with oscillating potentials is presented, where the main topics are: (1) the essential self adjointness of higher-order elliptic operators, (2) the characterization of the domain for nonnegative potentials, and (3) the m-accretivity and m-dispersiveness of degenerate-elliptic operators of second order in L P (R m ).
Abstract: This will be a partial survey, with some new results included, of recent results on (essential) selfadjointness problems and their generalizations for linear differential operators. The main topics will be: the (essential) selfadjointness of second-order elliptic operators with oscillating potentials; the (essential) selfadjointness of higher-order elliptic operators; characterization of the domain for nonnegative potentials; the m-accretivity and m-dispersiveness of degenerate-elliptic operators of second order in L P (R m ).
46 citations
1 Jan 1981
TL;DR: In this paper, the Riemann integral of vector-valued and operator-valued functions is discussed and a characterization of isometric and unitary operators is given, with L 2-spaces as examples.
Abstract: In Section 1.1 we give the definition and some elementary properties of a separable Hilbert space, with L2-spaces as examples. Section 1.2 contains the basic notions about linear operators and a characterization of isometric and unitary operators. Section 1.3 is devoted to a discussion of the Riemann integral of vector-valued and operator-valued functions.
Book•
1 Jun 1981
TL;DR: In this article, a C*-Algebra approach to the Cowen-Douglas Theory is presented, which is based on periodic distribution groups and isomorphisms of Automorphism Groups of Type II factors.
Abstract: On Closed Operator Algebras Generated by Analytic Functional Calculi.- A Conjecture Concerning the Pure States of B(H) and a Related Theorem.- A C*-Algebra Approach to the Cowen-Douglas Theory.- On Periodic Distribution Groups.- On the Smoothness of Elements of Ext.- Triviality Theorems for Hilbert Modules.- Exact Controllability and Spectrum Assignment.- Generalized Derivations.- Commutants Modulo the Compact Operators of Certain CSL Algebras.- Similarity of Operator Blocks and Canonical Forms. II. Infinite Dimensional Case and Wiener-Hopf Factorization.- Unitary Orbits of Power Partial Isometries and Approximation by Block-Diagonal Nilpotents.- Isomorphisms of Automorphism Groups of Type II Factors.- A Spectral Residuum for Each Closed Operator.- Two Applications of Hankel Operators.- A Rohlin Type Theorem for Groups Acting on von Neumann Algebras.- Derivations of C*-Algebras which Are Invariant Under an Automorphism Group.- Remarks on Ideals of the Calkin-Algebra for Certain Singular Extensions.- Modelling by L2-Bounded Analytic Functions.- The Maximal Function of Doubly Commuting Contractions.- Remarks on Hilbert-Schmidt Perturbations of Almost - Normal Operators.- Derivation Ranges: Open Problems.
TL;DR: In this article, a generalized Fourier transform representing the absolutely continuous part of the Schrodinger operator as multiplication by ¦ξ¦ 2 in the asymptotic cone is constructed.
TL;DR: In this article, a model representation similar to the Nagy-Foias model for dissipative operators is presented, which allows us to calculate the action of the resolvent of a nondissipative operator on selected subspaces.
Abstract: In the paper one investigates nondissipative operators L in a Hilbert space; for them one constructs a model representation similar to the Nagy-Foias model for dissipative operators. In this representation one succeeds to calculate the action of the resolvent of a nondissipative operator on selected subspaces. This allows us to relinquish the consideration of the
-self-adjoint dilation of the operator, whose spectral representation involves considerable difficulties. Isolated results are new also for the dissipative case which is not excluded. In part I one considers the “triangular” factorization of the characteristic funcion of the operator L and one carries out the proof of the fundamental theorem which gives a formula for the calculation of (L-No)−1(JmNoO TmNo
TL;DR: In this article, a new sufficient condition called maximal monotonicity is defined, which guarantees that the sum of two maximal monotone operators is again maximal-monotone.
Abstract: : A wide variety of problems involving nonlinear partial differential equations, subject to boundary conditions, may be shown to have solutions by establishing that the associated differential operators satisfy a certain technical condition This condition, called maximal monotonicity, allows the use of a well developed abstract theory which includes results about existence, regularity, etc, of solutions of operator equations It is quite useful, therefore, to have easily verified conditions which imply that an operator is maximal monotone Frequently an operator may be regarded as the sum of simpler components This paper gives a new sufficient condition which guarantees that the sum of two maximal monotone operators is again maximal monotone (Author)
TL;DR: In this article, a generalization of generalized inverses has been proposed for the problem of finding least-squares solutions in a linear operator with finite ascent and descent, where the operator is not necessarily one-to-one or onto.
TL;DR: In this article, the authors studied the relation between equicontinuity and collective compactness of continuous linear operators and derived a result on factoring compact sets of compact operators compactly and uniformly through one and the same reflexive Banach space.
TL;DR: In this article, a Parametrix constructions for some classes of right-invariant differential operators on the heisenberg group are given for partial differential equations on the group.
Abstract: (1981). Parametrix constructions for some classes of right-invariant differential operators on the heisenberg group. Communications in Partial Differential Equations: Vol. 6, No. 12, pp. 1363-1405.
TL;DR: In this paper, the essential spectrum and index function of the operator X -* AXB, where A, B, and X are Hilbert space operators, is described and analogous results for the restriction of this operator to a norm ideal and partial analogues are given for sums of such operators for the case when the operators act on a Banach space.
Abstract: This paper describes the essential spectrum and index function of the operator X -* AXB, where A, B, and X are Hilbert space operators. Analogous results are given for the restriction of this operator to a norm ideal and partial analogues are given for sums of such operators and for the case when the operators act on a Banach space.
TL;DR: In this article, linear factorization relations are derived for the matrix elements of quantum mechanical operators defined on some space H = H1⊕⋅2 which are diagonalizable on H1.
Abstract: Linear factorization relations are derived for the matrix elements of quantum mechanical operators defined on some space H = H1⊕⋅2 which are diagonalizable on H1. The coefficients in these relationships do not depend on the operators per se but do depend on the representations in which the operators are diagonal. The formulation is very general with regard to the nature of the ’’input’’ information in the factorization. With each choice of input information there are associated consistency conditions. The consistency conditions, in turn, give rise to a flexibility in the form of the factorization relations. These relations are examined in detail for the operators of scattering theory which are local in the internal molecular coordinates. In particular, this includes S and T matrices in the energy sudden (ES) approximation. A similar development is given for the square of the magnitude of operator matrix elements appropriately averaged over ’’symmetry classes.’’ In the ES these relations apply to transitio...
TL;DR: In this paper, the authors extended the Subdifferential of composite functions to the framework of ordered topological vector spaces by using the sandwich theorem for convex operators, derived from an extension of the Hahn-Banach theorem.
TL;DR: The operations of diffraction, phase conjugation, and the rigid motions in the plane are shown to be unitary operators on the space of all the band-limited fields whose diffracted wave fields are homogeneous.
Abstract: The operations of diffraction, phase conjugation, and the rigid motions in the plane are shown to be unitary operators on the space of all the band-limited fields whose diffracted wave fields are homogeneous. Commutation relations are given for these operators that explain many of the symmetries of such wave fields. Some of these symmetries are known, and some have not been mentioned before.
TL;DR: In this paper, the spectral theory of operators in Banach spaces is employed to treat a class of degenerate evolution equations, where a basic role is played by the assumption that the Banach space under consideration may be expressed as a direct sum of two suitable subspaces.
TL;DR: In this article, an algebra of tensors of order four which depend on a unit vector and the Kronecker delta is investigated, and it is shown that the algebra has natural applications in classical elasticity as well as in the continuum theory of dislocations.