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 a solution to the Phillips-Kato restricted extension problem about description and parametrization of the domains of all maximal accretive and sectorial quasi-self-adjoint extensions of a closed, densely defined nonnegative operator S in some Hilbert space.
Abstract: We present the solution to the Phillips–Kato restricted extension problem about description and parametrization of the domains of all maximal accretive and sectorial quasi-self-adjoint extensions \({\widetilde S (S\subset \widetilde S\subset S^*)}\) of a closed, densely defined nonnegative operator S in some Hilbert space. This description and parametrization are presented in terms of some sort of an analogy of von Neumann’s formulas for quasi-self-adjoint extensions. We use the approach proposed by Arlinskiĭ and Tsekanovskiĭ (Integr Equ Oper Theory 51:319–356, 2005) and our new formulas match the corresponding ones in the case of nonnegative self-adjoint extensions of S. An application to operators corresponding to finite number δ′-interactions on the real line is given as well as to the parametrization of all resolvents of maximal accretive extensions.
15 citations
Cites background from "On proper accretive extensions of p..."
...We need the followings results established in [6]....
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TL;DR: In this paper, a method of determining the proper dissipative extensions of a dual pair of operators on a Hilbert space is presented. But the stability of the numerical range of the different extensions is not investigated.
Abstract: Let A and $${(-\widetilde{A})}$$
be dissipative operators on a Hilbert space $${\mathcal{H}}$$
and let $${(A,\widetilde{A})}$$
form a dual pair, i.e. $${A \subset \widetilde{A}^*}$$
, resp. $${\widetilde{A} \subset A^*}$$
. We present a method of determining the proper dissipative extensions $${\widehat{A}}$$
of this dual pair, i.e. $${A\subset \widehat{A} \subset\widetilde{A}^*}$$
provided that $${\mathcal{D}(A)\cap\mathcal{D}(\widetilde{A})}$$
is dense in $${\mathcal{H}}$$
. Applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed. Finally, we investigate the stability of the numerical range of the different dissipative extensions.
9 citations
TL;DR: In this article, a method of determining the proper dissipative extensions of a dual pair of dissipative operators on a Hilbert space is presented. But the stability of the numerical range of the different extensions is not investigated.
Abstract: Let A and \({(-\widetilde{A})}\) be dissipative operators on a Hilbert space \({\mathcal{H}}\) and let \({(A,\widetilde{A})}\) form a dual pair, i.e. \({A \subset \widetilde{A}^*}\), resp. \({\widetilde{A} \subset A^*}\). We present a method of determining the proper dissipative extensions \({\widehat{A}}\) of this dual pair, i.e. \({A\subset \widehat{A} \subset\widetilde{A}^*}\) provided that \({\mathcal{D}(A)\cap\mathcal{D}(\widetilde{A})}\) is dense in \({\mathcal{H}}\). Applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed. Finally, we investigate the stability of the numerical range of the different dissipative extensions.
8 citations
Cites methods from "On proper accretive extensions of p..."
...Besides the classical results of von Neumann on the theory of selfadjoint extensions of a given symmetric operator [36] and of Krĕın, Birman, Vishik and Grubb on positive selfadjoint and maximally sectorial extensions of a given symmetric operator with positive numerical range [29, 41, 11, 26, 1, 2], let us also mention the results of authors like Arlinskĭı, Belyi, Derkach, Kovalev, Malamud, Mogilevskii and Tsekanovskĭı [4, 6, 7, 8, 17, 18, 33, 34, 35, 39, 40] who have made many contributions using form methods and boundary triples in order to determine maximally sectorial and maximally accretive extensions of a given sectorial operator.1.2 Let us also mention examples, where explicit computations of maximally dissipative (resp. accretive) extensions for positive symmetric differential operators [21], [22] and for sectorial Sturm-Liouville operators [13] have been made....
[...]
...Besides the classical results of von Neumann on the theory of selfadjoint extensions of a given symmetric operator [36] and of Krĕın, Birman, Vishik and Grubb on positive selfadjoint and maximally sectorial extensions of a given symmetric operator with positive numerical range [29, 41, 11, 26, 1, 2], let us also mention the results of authors like Arlinskĭı, Belyi, Derkach, Kovalev, Malamud, Mogilevskii and Tsekanovskĭı [4, 6, 7, 8, 17, 18, 33, 34, 35, 39, 40] who have made many contributions using form methods and boundary triples in order to determine maximally sectorial and maximally accretive extensions of a given sectorial operator....
[...]
01 Jan 2000
TL;DR: For a closed densely defined sectorial and coercive operator, a description of all maximal accretive extensions is given in this article, and applications to a second order partial elliptic differential operator are given.
Abstract: For a closed densely defined sectorial and coercive operator a description of all maximal accretive extensions is obtained. Applications to a second order partial elliptic differential operator are given.
5 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....
[...]
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