Catálogo de publicaciones - libros
Operator Theory in Inner Product Spaces
Karl-Heinz Förster ; Peter Jonas ; Heinz Langer ; Carsten Trunk (eds.)
Resumen/Descripción – provisto por la editorial
No disponible.
Palabras clave – provistas por la editorial
Operator Theory; Functions of a Complex Variable; Functional Analysis
Disponibilidad
Institución detectada | Año de publicación | Navegá | Descargá | Solicitá |
---|---|---|---|---|
No detectada | 2007 | SpringerLink |
Información
Tipo de recurso:
libros
ISBN impreso
978-3-7643-8269-8
ISBN electrónico
978-3-7643-8270-4
Editor responsable
Springer Nature
País de edición
Reino Unido
Fecha de publicación
2007
Información sobre derechos de publicación
© Birkhäuser Verlag AG 2007
Cobertura temática
Tabla de contenidos
Linear Operators in Almost Krein Spaces
Tomas Ya. Azizov; Lioudmila I. Soukhotcheva
The aim of this paper is to study the completeness and basicity problems for selfadjoint operators of the class K(H) in almost Krein spaces and prove criteria for the basicity and completeness of root vectors of linear pencils.
Palabras clave: Krein space; operator pencil; completeness and basicity problem.
Pp. 1-11
Generalized Resolvents of a Class of Symmetric Operators in Krein Spaces
Jussi Behrndt; Annemarie Luger; Carsten Trunk
Let A be a closed symmetric operator of defect one in a Krein space K and assume that A possesses a self-adjoint extension in K which locally has the same spectral properties as a definitizable operator. We show that the Krein-Naimark formula establishes a bijective correspondence between the compressed resolvents of locally definitizable self-adjoint extensions à of A acting in Krein spaces K x H and a special subclass of meromorphic functions.
Palabras clave: Generalized resolvents; Krein-Naimark formula; self-adjoint extensions; locally definitizable operators; locally definitizable functions; boundary value spaces; Weyl functions; Krein spaces.
Pp. 13-32
Block Operator Matrices, Optical Potentials, Trace Class Perturbations and Scattering
Jussi Behrndt; Hagen Neidhardt; Joachim Rehberg
For an operator-valued block-matrix model, which is called in quantum physics a Feshbach decomposition, a scattering theory is considered. Under trace class perturbations the channel scattering matrices are calculated. Using Feshbach’s optical potential it is shown that for a given spectral parameter the channel scattering matrices can be recovered either from a dissipative or from a Lax-Phillips scattering theory.
Palabras clave: Feshbach decomposition; optical potential; Lax-Phillips scattering theory; dissipative scattering theory; scattering matrix; characteristic function; dissipative operators.
Pp. 33-49
Asymptotic Expansions of Generalized Nevanlinna Functions and their Spectral Properties
Vladimir Derkach; Seppo Hassi; Henk de Snoo
Asymptotic expansions of generalized Nevanlinna functions Q are investigated by means of a factorization model involving a part of the generalized zeros and poles of nonpositive type of the function Q . The main results in this paper arise from the explicit construction of maximal Jordan chains in the root subspace R_∞( S _F) of the so-called generalized Friedrichs extension. A classification of maximal Jordan chains is introduced and studied in analytical terms by establishing the connections to the appropriate asymptotic expansions. This approach results in various new analytic characterizations of the spectral properties of selfadjoint relations in Pontryagin spaces and, conversely, translates analytic and asymptotic properties of generalized Nevanlinna functions into the spectral theoretical properties of self-adjoint relations in Pontryagin spaces.
Palabras clave: Generalized Nevanlinna function; asymptotic expansion; Pontryagin space; symmetric operator; selfadjoint extension; operator model; factorization; generalized Friedrichs extension.
Pp. 51-88
A Necessary Aspect of the Generalized Beals Condition for the Riesz Basis Property of Indefinite Sturm-Liouville Problems
Andreas Fleige
For the Sturm-Liouville eigenvalue problem − f ″ = γrf [−1, 1] with Dirichlet boundary conditions and with an indefinite weight function r changing it’s sign at 0 we discuss the question whether the eigenfunctions form a Riesz basis of the Hilbert space L _| r | ^2 [−1, 1]. In the nineties the sufficient so called generalized one hand Beals condition was found for this Riesz basis property. Now using a new criterion of Parfyonov we show that already the old approach gives rise to a necessary and sufficient condition for the Riesz basis property under certain additional assumptions.
Palabras clave: Indefinite Sturm-Liouville problem; Riesz basis; definitizable operator.
Pp. 89-94
On Reducible Nonmonic Matrix Polynomials with General and Nonnegative Coefficients
K. -H. Förster; B. Nagy
We consider nonmonic quadratic polynomials acting on a general or on a finite-dimensional linear space as a continuation of our work in [ 7 , 8 ]. Conditions are given for the existence of right roots, if the coefficient operators have lower block triangular representations. In the finite-dimensional case we consider (in a certain sense, entrywise) nonnegative coefficient matrices in the general (reducible) case, and extend several earlier results from the case of irreducible coefficients. In particular, we generalize results of Gail, Hantler and Taylor [ 9 ]. We show that our general methods are sufficiently strong to prove a remarkable result by Butler, Johnson and Wolkowicz [ 3 ], proved there by ingenious ad hoc methods.
Palabras clave: Reducible matrix polynomials; block triangular operator coefficients; (entrywise) nonnegative matrix coefficients; nonnegative matrix roots.
Pp. 95-109
On Exceptional Extensions Close to the Generalized Friedrichs Extension of Symmetric Operators
Seppo Hassi; Henk de Snoo; Henrik Winkler
If the Q -function Q corresponding to a closed symmetric operator S with defect numbers (1, 1) and one of its selfadjoint extensions belongs to the Kac class N _1 then it is known that all except one of the Q -functions of S belong to N _1, too. In this note the situation that the given Q -function does not belong to the class N _1 is considered. If Q ∈ N _p, i.e., if the restriction of the spectral measure of Q on the positive or the negative axis corresponds to an N _1-function, then Q itself is the Q -function of the exceptional extension, and, hence, it is associated with the generalized Friedrichs extension of S . If Q or, equivalently, the spectral measure of Q is symmetric, or if the difference of Q and a symmetric Nevanlinna function belongs to the class N _1 or N _p, then Q is still exceptional in a wider sense. Similar results hold for the generalized Kre i n-von Neumann extension of the symmetric operator.
Palabras clave: -function; generalized Friedrichs extension; generalized Kren-von Neumann extension; Kac class.
Pp. 111-120
On the Spectrum of the Self-adjoint Extensions of a Nonnegative Linear Relation of Defect One in a Krein Space
P. Jonas; H. Langer
A nonnegative symmetric linear relation A _0 with defect one in a Krein space H has self-adjoint extensions which are not nonnegative. If the resolvent set of such an extension A is not empty, A has a so-called exceptional eigenvalue α . For α ≠ 0, ∞ this means that α is an eigenvalue in the open upper half-plane, or a positive eigenvalue with a nonpositive eigenvector, or a negative eigenvalue with a nonnegative eigenvector. In this paper we study these exceptional eigenvalues and their dependence on a parameter if the selfadjoint extensions of A _0 are parametrized according to M. G. Krein’s resolvent formula. An essential tool is a family of generalized Nevanlinna functions of the class N _1 and their zeros or generalized zeros of nonpositive type.
Palabras clave: Linear relations in Krein spaces; nonnegative operators; operators with one negative square; selfadjoint extensions; generalized Nevanlinna functions.
Pp. 121-158
Canonical Differential Equations of Hilbert-Schmidt Type
Michael Kaltenbäck; Harald Woracek
A canonical system of differential equations, or Hamiltonian system, is a system of order two of the form Jy′ ( x ) = −zH ( x ) y ( x ), x ∈ ℝ^+. We characterize the property that the selfadjoint operators associated to a canonical system have resolvents of Hilbert-Schmidt type in terms of the Hamiltonian H as well as in terms of the associated Titchmarsh-Weyl coefficient.
Palabras clave: Canonical differential equation; Hilbert-Schmidt.
Pp. 159-168
Spectral Analysis of Differential Operators with Indefinite Weights and a Local Point Interaction
Ilia Karabash; Aleksey Kostenko
We consider quasi-self-adjoint extensions of the symmetric operator $$ A = - (\operatorname{sgn} x)\frac{{d^2 }} {{dx^2 }},dom(A) = \{ f \in W_2^2 (\mathbb{R}):f(0) = f'(0) = 0\} $$ , in the Hilbert space L ^2(ℝ). The main result is a criterion of similarity to a normal operator for operators of this class. The spectra and resolvents of these extensions are described. As an application we describe the main spectral properties of the operators $$ (\operatorname{sgn} x)\left( { - \tfrac{{d^2 }} {{dx^2 }} + c\delta } \right)and (\operatorname{sgn} x)\left( { - \tfrac{{d^2 }} {{dx^2 }} + c\delta '} \right) $$ .
Palabras clave: Symmetric operator; quasi-self-adjoint extensions; similarity problem; boundary triplets; Weyl functions.
Pp. 169-191