Showing papers on "Multiplication operator published in 1970"
TL;DR: In this article, a new formalism, termed the resolution space, is presented within which the theory of causal systems may be unified and extended, which allows the various aspects of network and system theory which are dependent on the time parameter to be studied in operator theoretic context without the detailed structure of a function space.
Abstract: A new formalism, termed the resolution space, is presented within which the theory of causal systems may be unified and extended. The resulting formalism, which is defined as a Hilbert space together with a resolution of the identity, readily includes the commonly encountered function and sequence space causality concepts yet is sufficiently straightforward to allow the various aspects of network and system theory which are dependent on the time parameter to be studied in an operator theoretic context without the detailed structure of a function space. Specific results include additive and multiplicative decomposition theorems for causal operators which naturally extend the “realizable part” and “spectral” decompositions of classical system theory and an integral representation theorem for linear operators on a resolution space. The general theory is illustrated with a number of examples concerning passive “networks”, those including an operator theoretic approach to the passive synthesis problem over an arbitrary resolution space.
57 citations
51 citations
43 citations
TL;DR: In this article, the authors consider extensions of a closed symmetric operator whose domain is not dense in the given Hilbert space and study self-adjoint extensions outside and the one-parameter families of operators generated by them in which are dissipative for.
Abstract: We consider extensions of a closed symmetric operator whose domain is, in general, not dense in the given Hilbert space . In particular, we study self-adjoint extensions outside and the one-parameter families of operators () generated by them in which are dissipative for . The set of all generalized resolvents of the operator is characterized.
42 citations
40 citations
TL;DR: A general formalism for defining an effective operator (≅ t ) in a truncated Hubert space is presented and is shown to depend on the operator v 21 which connects the included and the excluded Hilbert space as discussed by the authors.
34 citations
23 citations
CERN1
TL;DR: The operatorial expression for the gauge conditions is derived an explicit form for the twisting operator and the semi-twisting operator which takes theP states intoV states, and facilitates the proof of theV states factorization previously found by us.
Abstract: Using the operatorial expression for the gauge conditions, we derive an explicit form for the twisting operator and the semi-twisting operator which takes theP states intoV states. The expression for the last operator facilitates the proof of theV states factorization previously found by us. We show that the twisting operator proposed in the literature is inconsistent with multiple factorization,i.e. factorization of amplitudes with extermal spinning particles. The correct twisting operator depends on the integration variables of the twisted line, and automatically satisfies double-twist invariance.
22 citations
TL;DR: In this article, sufficient conditions for the self-adjointness of the Schrodinger operator in the whole space and in bounded regions were derived without any requirements concerning the existence of a spherically symmetric minorant of the potential satisfying the Titchmarsh-Sears conditions.
Abstract: Sufficient conditions are derived for the self-adjointness of the Schrodinger operator in the whole of space and in bounded regions, without supplementary boundary conditions and without any requirements concerning the existence of a spherically symmetric minorant of the potential satisfying the Titchmarsh-Sears conditions.
15 citations
TL;DR: In this article, sufficient conditions for bang-bang and singular optimal control are established in the case of linear operator equations with cost functions which are the sum of linear and quadratic terms, that is,Ax=u,J(u)=(r,x)+β(x,x), β>0.
Abstract: Sufficient conditions for bang-bang and singular optimal control are established in the case of linear operator equations with cost functionals which are the sum of linear and quadratic terms, that is,Ax=u,J(u)=(r,x)+β(x,x), β>0. For example, ifA is a bounded operator with a bounded inverse from a Hilbert spaceH into itself and the control setU is the unit ball inH, then an optimal control is bang-bang (has norm l) if 0⩽β 1/2∥A−1*r∥·∥A∥2.
01 Apr 1970
TL;DR: In this paper, it was shown that every nonnormal subnormal operator is the limit of a sequence of hyponormal and nonsubnormal operators, which is the same as the limit for a spectral set.
Abstract: An example is given of a nonnormal seminormal operator on a Hilbert space whose spectrum is thin (in the sense of von Neumann) and is therefore not a spectral set. It is shown that every nonnormal subnormal operator is the limit of a sequence of hyponormal and nonsubnormal operators.
TL;DR: In this paper, the existence of points of local instability of the Schmidt orthogonalization operator in a neighborhood of a fixed point has been proved, and the authors established the quadratic stability of this operator.
Abstract: In this article we prove the existence of points of local instability of the Schmidt orthogonalization operator and classify them. We establish the quadratic stability of this operator in a neighborhood of a fixed point.
TL;DR: In this article, the authors consider the properties of the spectrum of a differential operator derived from differential expressions of the fourth order and obtain an estimate of the number of eigenvalues lying in a given bounded interval of the real line.
Abstract: This paper considers the properties of the spectrum of a differential operator derived from differential expressions of the fourth order. With certain conditions on the coefficients of the differential expression the spectrum of the operator is discrete and an estimate is obtained of the number of eigenvalues lying in a given bounded interval of the real line. The results are compared with those obtained by alternative methods. Additional restrictions on the coefficients give special cases previously considered by other authors.