Abstract: 1. Functional Analysis. 2. Function Spaces. 3. Partial Differential Operators. 4. General L p Theory. 5. Relative Compactness. 6. Elliptic Operators. 7. Operators Bounded from Below. 8. Self-Adjoint Extensions. 9. Second Order Operators. 10. Applications. 11. Notes, Remarks and References. Bibliography. Subject Index.

Abstract: 1. Review of Basic Background.- 2. Linear Spaces and Operators.- 3. Canonical Forms.- 4. Quadratic Forms and Optimization.- 5. Differential and Difference Equations.- 6. Other Topics.- References.

Abstract: Publisher Summary This chapter presents second microlocalization and discusses the propagation of singularities for semi-linear hyperbolic equations. Fourier integral operators prove that second microlocalization can be made along any Lagrangean submanifold. Using invariance by Fourier integral operators, it is not difficult to define classes of operators mapping. Usual Fourier integral operators and singular diffeomorphisms are particular cases of a much larger class of transformations. They are well-behaved with respect to 2-microlocal calculus. Using invariance by usual Fourier integral operators, it is not difficult to define classes of various operators.

Abstract: with historical remarks.- General theory of derivations.- Noncommutative vectorfields.- Dissipations.- Additional remarks.

Abstract: The perturbative effects of the inter-electronic Coulomb interaction between atomic configurations can be reproduced by effective operators acting within a particular configuration under study. These effective operators may be resolved into orthogonal operators that act on N electrons at a time. For the atomic l shell, the symplectic labels associated with a given quasi-spin rank K can be found by referring to the branching rules for U(4l+2) to Sp(4l+2) even though the generators of U(4l+2) do not commute with the generators Q of the quasi-spin group SOQ(3). A given N-electron Hermitian orthogonal operator belonging to an irreducible representation of U(4l+2) is shown to correspond to even or odd K according to whether N is even or odd. For anti-Hermiticity, the connection is reversed. For the d shell, the groups U(10), Sp(10) and SO(5) are shown to provide an unambiguous classification of all operators of the type T(00)0, that is, for those scalar with respect to S, L and J. there are 1, 0, 4, 4, 12 and 0 operators corresponding to N=0, 1, 2, 3, 4 and 5. A complete tabulation of the matrix elements of these 21 operators is provided.

Abstract: This paper investigates the completion of the maximal Op*-algebra L+ (D) of (possibly) unbounded operators on a dense domain D in a Hilbert space. It is assumed that D is a Frechet space with respect to the graph topology. Let D+ denote the strong dual of D, equipped with the complex conjugate linear structure. It is shown that the completion of L+(D} (endowed with the uniform topology) is the space of continuous linear operators X (D, D+) . This space is studied as an ordered locally convex space with an involution and a partially defined multiplication. A characterizati on of bounded subsets of D in terms of self-adjoint operators is given. The existence of special factorizations for several kinds of operators is proved. It is shown that the bounded operators are uniformly dense in

Abstract: One of the principal results of the paper is that each scalar-type spectral operator in the quasicomplete locally convex space X is reflexive. The paper also studies in detail the relation between the theory of equicontinuous spectral measures in locally convex spaces and the order properties of equicontinuous Bade complete Boolean algebras of projections. o. Introduction. One of the principal results of this paper is that each scalar-type spectral operator T in the quasicomplete, locally convex space X is reflexive, i.e. the strongly closed subalgebra generated by the identity and T in .P(X), the space of continuous linear operators on X, consists precisely of those continuous linear operators on X which leave invariant each (closed) T-invariant subspace of X. For the case that X is a Banach space, this result was established by Gillespie [9] via an interesting factorization theorem in Banach function spaces, a method which does not appear to extend readily to the more general setting. The present approach, however, avoids factorization theorems by showing directly that each continuous linear functional on the strongly closed algebra generated by a Bade-complete, equicontinuous Boolean algebra of projections in X has a representation of the form < . x, x') for some x E X and x' E X', where X' denotes the dual space of X, a result which goes back to R. Pallu de la Barriere [20] for the case of Abelian von Neumann algebras in Hilbert space. Our method is based on ideas from the theory of Riesz spaces and yields considerable simplification of technique, even in the setting of Banach spaces. The cornerstone of the present paper is the extension of the reflexivity theorem of Bade [2] to the setting of locally convex spaces proved in [6] via the theory of closed spectral measures and further refined and sharpened in [5] using purely intrinsic methods, based on order considerations. One of the new features which emerged from the approach of [5] was a type of "automatic continuity" theorem for a certain class of everywhere defined linear operators, even in the absence of a suitable closed-graph theorem. This idea is exploited in §1 to show that an everywhere Received by the editors December 10, 1984. 1980 Mathematics Subject Classification. Primary 47D30, 47B40; Secondary 28A60, 46A40.

Abstract: We consider Lipschitz-continuous nonlinear maps in finite-dimensional Banach and Hilbert spaces. Boundedness and monotonicity of the operator are characterized quantitatively in terms of certain functionals. These functionals are used to assess qualitative properties such as invertibility, and also enable a generalization of some well-known matrix results directly to nonlinear operators. Closely related to the numerical range of a matrix, the Gerschgorin domain is introduced for nonlinear operators. This point set in the complex plane is always convex and contains the spectrum of the operator's Jacobian matrices. Finally, we focus on nonlinear operators in Hilbert space and hint at some generalizations of the von Neumann spectral theory.

Abstract: Canonical operator theory of paraxial optics is generalized to address the case of misaligned optics. The formal group structure is extended from the aligned case in terms of Heisenberg–Weil and inhomogeneous canonical transforms and the associated 3 × 3 augmented ray matrices. Certain misalignment phase shifts that are often mistreated and ignored have been derived and incorporated into the theory.

Abstract: An algebra of operators having the property of the title is constructed and it is used to give examples related to some recent invariant subspace results.

Abstract: We consider an iterative process in which one out of a finite set of possible operators is applied at each iteration. We obtain necessary and sufficient conditions for convergence to a common fixed point of these operators, when the order at which different operators are applied is left completely free, except for the requirement that each operator is applied infinitely many times. The theory developed is similar in spirit to Lyapunov stability theory. We also derive some very different, qualitatively, results for partially asynchronous iterative processes, that is for the case where certain requirements are imposed on the order at which the different operators are applied.

Abstract: For regular operators on a Banach lattice, we introduce and investigate two notions of order essential spectrum analogous to the essential spectrum and the Weyl spectrum for operators on Banach spaces. We also discuss related questions on the behaviour of the order spectrum under perturbation by r -compact operators.

Abstract: On definit la somme parallele d'operateurs non lineaires dans un espace de Hilbert a l'aide de leurs graphes

Abstract: The author studies pseudodifference operators on a discrete metric space, where the matrix elements of the operators decrease faster than a system of singular functions of the distance between points determining a matrix element. Similar estimates for matrix elements are proved for the inverse of a pseudodifference operator in the case where the weight functions increase faster than any function of the volume (the number of points in the ball of radius with prescribed center) and slower than the standard exponential function. Bibliography: 12 titles.

Abstract: Let H be a Krein space and U a unitary operator in H. Assume that for a certain open subset ¦£ of the unit circle T no point of Γ is accumulation point of the nonunitary spectrum σ(U)\T of U and that every point of Γ can be connected with 0 and ∞ by curves in the resolvent set ρ(U). Such an operator is called definitizable over Γ, if, roughly speaking, it has a spectral function over ¦£ (in the sense of the theory of definitizable operators). This class of operators, of course, contains the definitizable unitary operators in H. The main concern of this paper is to study some aspects of the behaviour of the spectral functions under compact perturbations of the corresponding operators within this class.