Some Applications of Operator-valued Herglotz Functions

Gesztesy, Fritz; Kalton, N. J.; Makarov, Konstantin A.; Tsekanovskiı̆, Eduard

doi:10.1007/978-3-0348-8247-7_13

Cited by 69 publications

(130 citation statements)

References 47 publications

(74 reference statements)

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“…Our principal motivation in studying this problem stems from our general interest in operator-valued Herglotz functions (cf. [12], [37], [38], [39], [40], [41], [44], [85]) and its possible applications in the areas of inverse spectral theory and completely integrable systems. More precisely, using higher-order expansions of the type (1.1), one can prove trace formulas for Q(x) and certain higher-order differential polynomials in Q(x), following an approach pioneered in [42] (see also [35], [36]).…”

Section: Introductionmentioning

confidence: 99%

Weyl-Titchmarsh M -Function Asymptotics for Matrix-valued Schrödinger Operators

Clark

Gesztesy

2001

Proceedings of the London Mathematical Society

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Section: Introductionmentioning

confidence: 99%

Weyl-Titchmarsh M -Function Asymptotics for Matrix-valued Schrödinger Operators

Clark

Gesztesy

2001

Proceedings of the London Mathematical Society

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“…The description of L 2 (Σ, H) in Theorem 2.14 allows us to give a consistent proof of Proposition 5.2 in [18]. In this connection, we mention the recent paper [15], in which a different model of a symmetric operator was studied with the aid of Herglotz functions.…”

Section: §1 Introductionmentioning

confidence: 82%

Spectral theory of operator measures in Hilbert space

Malamud

2004

St. Petersburg Math. J.

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Abstract. In §2 the spaces L 2 (Σ, H) are described; this is a solution of a problem posed by M. G. Kreȋn.In §3 unitary dilations are used to illustrate the techniques of operator measures. In particular, a simple proof of the Naȋmark dilation theorem is presented, together with an explicit construction of a resolution of the identity. In §4, the multiplicity function N Σ is introduced for an arbitrary (nonorthogonal) operator measure in H. The description of L 2 (Σ, H) is employed to show that this notion is well defined. As a supplement to the Naȋmark dilation theorem, a criterion is found for an orthogonal measure E to be unitarily equivalent to the minimal (orthogonal) dilation of the measure Σ.In §5 it is proved that the set Ω Σ of all principal vectors of an arbitrary operator measure Σ in H is massive, i.e., it is a dense G δ -set in H. In particular, it is shown that the set of principal vectors of a selfadjoint operator is massive in any cyclic subspace.In §6, the Hellinger types are introduced for an arbitrary operator measure; it is proved that subspaces realizing these types exist and form a massive set.In §7, a model of a symmetric operator in the space L 2 (Σ, H) is studied.

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“…Some additional properties of the Weyl-Titchmarsh functions and their applications can be found in [4], [5].…”

Section: [(H −Zi)d(h)]mentioning

confidence: 99%

“…In this paper we study a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function M H,H (z) generated by the densely defined symmetric operator H and its self-adjoint extension H acting on some Hilbert space H [3], [4], [5]. The new properties of these functions as well as a new version of the functional model for the pair (H, H) in terms of M H,H (z) are obtained.…”

Section: Introductionmentioning

confidence: 99%

On periodic matrix-valued Weyl?Titchmarsh functions

Bekker

2004

Journal of Mathematical Analysis and Applications

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We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl -Titchmarsh matrixvalued functions as well as a new version of the functional model in such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its self-adjoint extension generating periodic Weyl-Titchmarsh matrix-valued function. We study pairs of operators (a symmetric operator and its self-adjoint extension) with constant Weyl-Titchmarsh matrix-valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in Weyl's form. As a consequence of such an approach we obtain the Stone-von Neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups. Our examples include multiplication operators in weighted spaces, first and second order differential operators, as well as the Schrödinger operator with linear potential and its perturbation by bounded periodic potential.

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Some Applications of Operator-valued Herglotz Functions

Cited by 69 publications

References 47 publications

Weyl-Titchmarsh M -Function Asymptotics for Matrix-valued Schrödinger Operators

Weyl-Titchmarsh M -Function Asymptotics for Matrix-valued Schrödinger Operators

Spectral theory of operator measures in Hilbert space

On periodic matrix-valued Weyl?Titchmarsh functions

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