The Theory of Bi-Extensions of Operators on Rigged Hilbert Spaces. Unbounded Operator Colligations and Characteristic Functions

In this paper we provide some additional results related to Krein's resolvent formula for a non-densely defined symmetric operator. We show that coefficients in Krein's formula can be expressed in terms of analogues of the classical von Neumann formulas. The relationship between two Weyl-Tichmarsh m-functions corresponding to self-adjoint extensions of a non-densely defined symmetric operator is established.  2001 Elsevier Science

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“…It is easy to see that λ = λ ∩ 0 , and λ is the set of solutions of the equationȦ * g = λg (see [23]), whereȦ * → 0 is the adjoint operator toȦ.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…The analogues of the von Neumann formulas for the non-dense case, also known as Krasnoselskii's formulas [14], take a form of indirect decompositions. Besides the Krasnoselskii formulas there is another analogue of the von Neumann formulas obtained by Arlinskii,Smuljan, and Tsekanovskii [4,23], where these decompositions are direct.…”

Section: Introductionmentioning

confidence: 95%

On Krein's Formula in the Case of Non-densely Defined Symmetric Operators

Belyi

Menon

Tsekanovskiı̆³

2001

Journal of Mathematical Analysis and Applications

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“…We complete this section with a glance at previous work. Early work on conservative systems dates back to Brodskiȋ [6], [7], Helton [11], Livšic [13], Livšic and Yantsevich [14], Sz.-Nagy and Foiaş [27], and Tsekanovski and Smuljan [28], to mention only a few. Most of these references consider discrete-time systems, where the difficulties are considerably smaller.…”

Section: Jarmo Malinen Olof J Staffans and George Weissmentioning

confidence: 99%

When is a linear system conservative?

2006

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Abstract. We consider infinite-dimensional linear systems without a-priori wellposedness assumptions, in a framework based on the works of M. Livšic, M. S. Brodskiȋ, Y. L. Smuljan, and others. We define the energy in the system as the norm of the state squared (other, possibly indefinite quadratic forms will also be considered). We derive a number of equivalent conditions for a linear system to be energy preserving and hence, in particular, well posed. Similarly, we derive equivalent conditions for a system to be conservative, which means that both the system and its dual are energy preserving. For systems whose control operator is one-to-one and whose observation operator has dense range, the equivalent conditions for being conservative become simpler, and reduce to three algebraic equations. Introduction.In this paper we give a number of algebraic characterizations of energy preserving and of conservative linear systems in terms of the operators appearing in a differential/algebraic state space description of the system. Such algebraic characterizations may be useful for recognizing that certain linear partial differential or delay equations describe energy preserving or conservative systems. Our conditions do not include an a-priori well-posedness assumption, but well-posedness follows if the system is energy preserving.

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“…3], [45], [46,Ch. 13], [49]- [51], [55], [58], [61], [65], [66], [71], [72], [73], [75]- [77], [82] and the references cited therein.…”

Section: Ac([0 R]) Denotes the Set Of Absolutely Continuous Functionmentioning

confidence: 99%

“…3], [45], [46,Ch. 13], [49], [50], [51], [53], [54], [55], [58], [61], [63], [64], [65], [66], [67], [69], [70], [71], [72], [73], [75], [74], [77], [78], [79], [82], and the references cited therein. We start by explicitly computing operators of the type…”

Section: Krein-type Resolvent Formulasmentioning

confidence: 99%

Boundary Data Maps for Schrödinger Operators on a Compact Interval

Clark

Gesztesy

Mitrea

2010

Math. Model. Nat. Phenom.

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Abstract. We provide a systematic study of boundary data maps, that is, 2 × 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrödinger operators on a compact interval [0, R] with separated boundary conditions at 0 and R. Most of our results are formulated in the non-self-adjoint context.Our principal results include explicit representations of these boundary data maps in terms of the resolvent of the underlying Schrödinger operator and the associated boundary trace maps, Krein-type resolvent formulas relating Schrödinger operators corresponding to different (separated) boundary conditions, and a derivation of the Herglotz property of boundary data maps (up to right multiplication by an appropriate diagonal matrix) in the special self-adjoint case.

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The Theory of Bi-Extensions of Operators on Rigged Hilbert Spaces. Unbounded Operator Colligations and Characteristic Functions

Cited by 50 publications

References 15 publications

On Krein's Formula in the Case of Non-densely Defined Symmetric Operators

On Krein's Formula in the Case of Non-densely Defined Symmetric Operators

When is a linear system conservative?

Boundary Data Maps for Schrödinger Operators on a Compact Interval

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