scispace - formally typeset
Search or ask a question
Journal ArticleDOI

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.

Content maybe subject to copyright    Report






Citations
More filters
Journal ArticleDOI
TL;DR: In this paper, a description of all maximal accretive extensions and their resolvents for a densely defined closed sectorial operator in terms of abstract boundary conditions is given.
Abstract: A description of all the maximal accretive extensions and their resolvents is given for a densely defined closed sectorial operator in terms of abstract boundary conditions. These results are applied to parametrize all the m-accretive extensions of a symmetric operator in a planar model of one-centre point interaction. Bibliography: 40 titles.

4 citations

01 Jan 2010
TL;DR: In this article, a parametrization of all quasi-self-adj oint maximal accretive extensions and maximal sectorial extensions (with vertex at 0 z = and the acute semi-angle α ) for a densely defined closed nonnegative symmetric operator is given.
Abstract: Summary. We give a new parametrization of all quasi-self-adj oint maximal accretive extensions and maximal sectorial extensions (with vertex at 0 z = and the acute semi-angle α ) for a densely defined closed nonnegative symmetric operator. An applicati on to point interactions model on the real line is considered.

2 citations

Posted Content
TL;DR: In this paper, a necessary and suffcient condition for any extension of a dissipative operator with domain contained in the domain of the domain is presented. But this condition is not applicable to all dissipative operators.
Abstract: We consider dissipative operators $A$ of the form $A=S+iV$, where both $S$ and $V\geq 0$ are assumed to be symmetric but neither of them needs to be (essentially) selfadjoint. After a brief discussion of the relation of the operators $S\pm iV $to dual pairs with the so called common core property, we present a necessary and suffcient condition for any extension of $A$ with domain contained in $\mathcal{D}((S-iV)^*)$ to be dissipative. We will discuss several special situations in which this condition can be expressed in a particularly nice form -- accessible to direct computations. Examples involving ordinary differential operators are given.

2 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider sectorial operators with symmetric symmetric operators and give a criterion for when the quadratic form of the self-adjoint operator is closed.
Abstract: We consider densely defined sectorial operators $$A_\pm $$ that can be written in the form $$A_\pm =\pm iS+V$$ with $$\mathcal {D}(A_\pm )=\mathcal {D}(S)=\mathcal {D}(V)$$ , where both S and $$V\ge \varepsilon >0$$ are assumed to be symmetric. We develop an analog to the Birmin–Kreĭn–Vishik–Grubb (BKVG) theory of selfadjoint extensions of a given strictly positive symmetric operator, where we will construct all maximally accretive extensions $$A_D$$ of $$A_+$$ with the property that $$\overline{A_+}\subset A_D\subset A_-^*$$ . Here, D is an auxiliary operator from $$\ker (A_-^*)$$ to $$\ker (A_+^*)$$ that parametrizes the different extensions $$A_D$$ . After this, we will give a criterion for when the quadratic form $$\psi \mapsto {\text{ Re }}\langle \psi ,A_D\psi \rangle $$ is closable and show that the selfadjoint operator $$\widehat{V}$$ that corresponds to the closure is an extension of V. We will show how $$\widehat{V}$$ depends on D, which—using the classical BKVG-theory of selfadjoint extensions—will allow us to define a partial order on the real parts of $$A_D$$ depending on D. All of our results will be presented in a way that emphasizes their connection to the classical BKVG-theory and that shows how the BKVG formulas generalize when considering sectorial operators $$A_\pm =\pm iS+V$$ instead of just a strictly positive symmetric operator V. Applications to second order ordinary differential operators are discussed.

2 citations

01 Jan 2016
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.

1 citations

References
More filters
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

Journal ArticleDOI

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....

    [...]

Journal ArticleDOI
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