On proper accretive extensions of positive linear relations
01 Jun 1995-Ukrainian Mathematical Journal (Kluwer Academic Publishers-Plenum Publishers)-Vol. 47, Iss: 6, pp 831-840
TL;DR: In this article, the authors established criteria for an accretive extension of a given positive symmetric linear relation to be proper and proved that such an extension is a proper extension.
Abstract: A linear relation S is called a proper extension of a symmetric linear relation S if S ⊂ S ⊂ S*. As is well known, an arbitrary dissipative extension of a symmetric linear relation is proper. In the present paper, we establish criteria for an accretive extension of a given positive symmetric linear relation to be proper.
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TL;DR: In this paper, the authors present an independent solution to von Neumann's problem on the parametrization in explicit form of all nonnegative self-adjoint extensions of a densely defined nonnegative symmetric operator.
Abstract: We develop a new approach and present an independent solution to von Neumann’s problem on the parametrization in explicit form of all nonnegative self-adjoint extensions of a densely defined nonnegative symmetric operator. Our formulas are based on the Friedrichs extension and also provide a description for closed sesquilinear forms associated with nonnegative self-adjoint extensions. All basic results of the well-known Krein and Birman-Vishik theory and its complementations are derived as consequences from our new formulas, including the parametrization (in the framework of von Neumann’s classical formulas) for all canonical resolvents of nonnegative selfadjoint extensions. As an application all nonnegative quantum Hamiltonians corresponding to point-interactions in \(\mathbb{R}^3\) are described.
54 citations
01 Jan 2009
TL;DR: In this paper, the authors consider the M Kreĭn classical papers on semi-bounded operators and the theory of contractive self-adjoint extensions of Hermitian contractions, and discuss their impact and role in the solution of J von Neumann's problem about parametrization in terms of his formulas of all nonnegative selfadjoint extension of nonnegative symmetric operators.
Abstract: We are going to consider the M Kreĭn classical papers on the theory of semi-bounded operators and the theory of contractive self-adjoint extensions of Hermitian contractions, and discuss their impact and role in the solution of J von Neumann’s problem about parametrization in terms of his formulas of all nonnegative self-adjoint extensions of nonnegative symmetric operators, in the solution of the Phillips-Kato extension problems (in restricted sense) about existence and parametrization of all proper sectorial (accretive) extensions of nonnegative operators, in bi-extension theory of non-negative operators with the exit into triplets of Hilbert spaces, in the theory of singular perturbations of nonnegative self-adjoint operators, in general realization problems (in system theory) of Stieltjes matrix-valued functions, in Nevanlinna-Pick system interpolation in the class of sectorial Stieltjes functions, in conservative systems theory with accretive main Schrodinger operator, in the theory of semi-bounded symmetric and self-adjoint operators invariant with respect to some groups of transformations New developments and applications to the singular differential operators are discussed as well
40 citations
TL;DR: In this article, a characterization for maximal monotone realizations for a certain class of nonlinear operators in terms of their corresponding boundary data spaces is provided, where the operators under consideration naturally arise in the study of evolutionary problems in mathematical physics.
31 citations
TL;DR: In this article, all closed sesquilinear forms associated with m-sectorial extensions of a densely defined sectorial operator with vertex at the origin are described, and the authors describe all closed m-sectorsial extensions associated with the m-veto operator.
Abstract: We describe all closed sesquilinear forms associated with m-sectorial extensions of a densely defined sectorial operator with vertex at the origin.
30 citations
Cites result from "On proper accretive extensions of p..."
...Note that if S is an m-sectorial extremal extension of S and DIS] = D[SN], then S = SN. It is proved in [ 24 ] that if S is a nonnegative symmetric operator, then the class ~,(S) coincides with the class of all proper m-sectorial extensions of S. Below, we show that a similar situation takes place in the general case where c~ ;e 0....
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01 Oct 2012
22 citations
References
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Book•
01 Jan 1966
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Abstract: "The monograph by T Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory
In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4) Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8) The fundamentals of semigroup theory are given in chapter 9 The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10
The first edition is now 30 years old The revised edition is 20 years old Nevertheless it is a standard textbook for the theory of linear operators It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field
Zentralblatt MATH, 836
19,846 citations
255 citations
"On proper accretive extensions of p..." refers background in this paper
...It is well known [ 2 ] that an arbitrary dissipative extension of a symmetric linear relation is proper....
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TL;DR: In this article, the authors prove that the shorted operator exists and develop various properties, including a relation to parallel addition, including the relation between the short operator and parallel addition.
Abstract: For a positive operator A acting on a Hilbert space, the shorted operator $\mathcal{L}( A )$ is defined to be the supremum of all positive operators which are smaller than A and which have range lying in a fixed subspace S. This maximization problem arises naturally in electrical network theory. In this paper we prove that the shorted operator exists, and develop various properties, including a relation to parallel addition [Anderson and Duffin, J. Math. Anal. Appl., 11 (1969), pp. 576–594]. The basic properties of the shorted operator were developed for finite-dimensional spaces by Anderson [this Journal, 20 (1971), pp. 520–525] ; some of these theorems remain true in all Hilbert spaces, but the proofs are different.
235 citations