Showing papers on "Nuclear operator published in 1978"
01 Jan 1978
106 citations
TL;DR: In this article, the weak and strong stabilizability problem of linear control systems on Hilbert space was studied and sufficient conditions for stabilization can be obtained by using a decomposition theorem in the structure theory of Hilbert space operators.
Abstract: This paper will study the weak and strong stabilizability problem of linear control systems on Hilbert space. It will be shown that sufficient conditions for stabilizability can be obtained by using a decomposition theorem in the structure theory of Hilbert space operators. The basic idea is "trivializing" the unitary "part" of a semigroup of bonded linear operators by means of a suitable "feedback" perturbation operator Controllability will not be involved in this process. However, it will be seen that further sufficient conditions as well as necessary conditions will be obtained with the aid of controllability. Extensions as well as limitatations of the familiar finite-dimensional results will also be discussed.
44 citations
40 citations
01 Jan 1978
TL;DR: In this article, it is shown that an operator A cannot be regarded as fully specified until its domain of definition (i.e., the set of those x in ℌ for which Ax is meaningful) has been specified.
Abstract: The idea of a linear operator or transformation in a Hilbert space ℌ (or a Banach space) is a direct generalization of the idea of a linear transformation in a finite-dimensional space. One point, however, needs emphasis (mainly because it is sometimes ignored, especially in books on quantum mechanics), namely, an operator A cannot be regarded as fully specified until its domain of definition (i.e., the set of those x in ℌ for which Ax is meaningful) has been specified; operators with different domains of definition have to be regarded as different operators. It is customary to require the domain of definition to be a linear set (manifold) in ℌ, for the obvious reason that if A is linear and Ax is defined in a set S, then Ay can be uniquely defined, by linearity, when y is any finite linear combination of elements of S. However, further extensions are not generally unique, except in special circumstances.
36 citations
TL;DR: In this article, the authors consider the problem of finding a strongly positive linear function on a *-algebra A of unbounded operators, under what conditions is every strongly positive function on A a trace functional, i.e. of the formf(a)=Trtta,a∈A, wheret is an appropriate positive nuclear operator.
Abstract: In this paper we consider the following problem: Given a *-algebraA of unbounded operators, under what conditions is every strongly positive linear functionalf onA a trace functional, i.e. of the formf(a)=Trtta,a∈A, wheret is an appropriate positive nuclear operator. Further, the linear functionalsf onA which can be represented asf(a)=Trta (f andt not necessarily positive) are characterized by their continuity in a certain topology. Some applications (canonical commutation relations on the Schwartz space, integrable representations of enveloping algebras) are discussed.
31 citations
26 citations
13 citations
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7 citations
TL;DR: In this article, the Fredholm determinant and related trace formulas for a class of operators which correspond to the restriction of integral operators with kernels of the form k(x,y) = χ(x)gv(x−y)+[1−χ (x)]fV(x+y) to the square |x|,|y| ⩽ T and evaluate the limit as T ↑ ∞, where χ denotes the indicator function of the right half-line [0,∞].
Abstract: In this paper we shall study the Fredholm determinant and related trace formulas for a class of operators which correspond to the restriction of integral operators with kernels of the form k(x,y) = χ(x)gv(x−y)+[1−χ(x)]fv(x−y) to the square |x|,|y| ⩽ T and shall evaluate the limit as T ↑ ∞ . Here χ denotes the indicator function of the right half-line [0,∞) . The results obtained generalize the well known formulas of M. Kac for the classical convolution operator in which g = f .
4 citations
TL;DR: In this article, the existence of wave operators for pairs of self adjoint operators acting in different Hilbert spaces has been studied for the Klein-Gordon equation, and a new criterion for wave operators has been proposed.
Abstract: We give a new criterion for the existence of wave operators for pairs of self adjoint operators acting in different Hilbert spaces. We apply this method to obtain results for the Klein-Gordon equation.
TL;DR: In this paper, a trace can be defined for operators belonging to a large class of majorizing and cone absolutely summing operators on a Banach lattice, and strongly measurable representations for a certain class of cone Absolutely Summing operators are derived.
Abstract: Operators T which are both majorizing and cone absolutely summing on a Banach lattice E are investigated. Compactness and nuclearity of such operators are discussed and it is shown that a trace can be defined for operators belonging to a large class of these operators. Special results are derived in the case where E is a Banach function space and T a kernel operator. Finally we derive strongly measurable representations for a certain class of cone absolutely summing operators thereby clarifying work done by Dinculeanu and J.J. Uhl.
TL;DR: In this article, it was shown that every compact quasinilpotent Hilbert space operator can be uniformly approximated by operators that are similar to the Volterra operator in L2([0, 1],dx).
Abstract: Every compact quasinilpotent Hilbert space operator can be uniformly approximated by operators that are similar to the Volterra operator in L2([0,1],dx).
01 Jan 1978
TL;DR: In this article, the authors dealt with several classes of bounded operators: compact, Hilbert-Schmidt, trace-class, and degenerate operators, and related them by the inclusions bounded ⊃ compact ⊆ Hilbert-schmidt ⊈ trace-classes ⊂ degenerate.
Abstract: This chapter deals with several classes of bounded operators: the compact, Hilbert-Schmidt, trace-class and degenerate operators; they are related by the inclusions bounded ⊃ compact ⊃ Hilbert-Schmidt ⊃ trace-class ⊃ degenerate.
TL;DR: In this paper, a characterization of operators T: X → Y, transforming bounded sets of the Banach space X into conditionally weakly compact sets of Y, and also a particular case where X = C(K) is considered.
Abstract: Certain classes of continuous linear operators in Banach and locally convex spaces are studied. A characterization of operators T: X → Y, transforming bounded sets of the Banach space X into conditionally weakly compact sets of the Banach space Y, is given, and also a particular case where X = C(K) is considered. It is proved that if E is a Frechet space and F is a complete (ℱ)-space, then the classes of absolutely summing and Nikodýmizing operators from E into F coincide.
TL;DR: In this paper, it was shown that a relevant part of the theory extends to bounded (in general nonlinear) operators in locally convex spaces, and that the theory can be extended to nonlinear operators in Banach spaces.