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Linear Operators in Almost Krein Spaces

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.
Tomas Ya. Azizov, Lioudmila I. Soukhotcheva

Generalized Resolvents of a Class of Symmetric Operators in Krein Spaces

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.
Jussi Behrndt, Annemarie Luger, Carsten Trunk

Block Operator Matrices, Optical Potentials, Trace Class Perturbations and Scattering

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.
Jussi Behrndt, Hagen Neidhardt, Joachim Rehberg

Asymptotic Expansions of Generalized Nevanlinna Functions and their Spectral Properties

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.
Vladimir Derkach, Seppo Hassi, Henk de Snoo

A Necessary Aspect of the Generalized Beals Condition for the Riesz Basis Property of Indefinite Sturm-Liouville Problems

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.
Andreas Fleige

On Reducible Nonmonic Matrix Polynomials with General and Nonnegative Coefficients

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.
K. -H. Förster, B. Nagy

On Exceptional Extensions Close to the Generalized Friedrichs Extension of Symmetric Operators

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 QN 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 Krein-von Neumann extension of the symmetric operator.
Seppo Hassi, Henk de Snoo, Henrik Winkler

On the Spectrum of the Self-adjoint Extensions of a Nonnegative Linear Relation of Defect One in a Krein Space

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.
P. Jonas, H. Langer

Canonical Differential Equations of Hilbert-Schmidt Type

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.
Michael Kaltenbäck, Harald Woracek

Spectral Analysis of Differential Operators with Indefinite Weights and a Local Point Interaction

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 \( (\operatorname{sgn} x)\left( { - \tfrac{{d^2 }} {{dx^2 }} + c\delta } \right)and (\operatorname{sgn} x)\left( { - \tfrac{{d^2 }} {{dx^2 }} + c\delta '} \right) \).
Ilia Karabash, Aleksey Kostenko

Normal Matrices in Degenerate Indefinite Inner Product Spaces

Complex matrices that are structured with respect to a possibly degenerate indefinite inner product are studied. Based on the theory of linear relations, the notion of an adjoint is introduced: the adjoint of a matrix is defined as a linear relation which is a matrix if and only if the inner product is nondegenerate. This notion is then used to give alternative definitions of selfadjoint and unitary matrices in degenerate inner product spaces and it is shown that those coincide with the definitions that have been used in the literature before. Finally, a new definition for normal matrices is given which allows the generalization of an extension result for positive invariant subspaces from the case of nondegenerate inner products to the case of degenerate inner products.
Christian Mehl, Carsten Trunk

Symmetric Hermite-Biehler Polynomials with Defect

The polynomial ω = P(λ) + iQ(λ) with real P(λ) and Q(λ) which belongs to Hermite-Biehler class (all its zeros lie in the open upper half-plane) and is symmetric \( (\omega ( - \bar \lambda ) = \overline {\omega (\lambda )} ) \) is modified as follows
$$ (\omega _c (\lambda ) = \tilde P(\lambda ^2 + c) + i\lambda \tilde \hat Q(\lambda ^2 + c), c > 0. $$
Here \( \tilde P(\lambda ^2 ) = P(\lambda ),\tilde \hat Q(\lambda ^2 ) = \lambda ^{ - 1} Q(\lambda ) \) and
$$ P(\lambda ) = \frac{{\omega (\lambda ) + \omega ( - \lambda )}} {2},Q(\lambda ) = \frac{{\omega (\lambda ) - \omega ( - \lambda )}} {{2i}} $$
The conditions are obtained necessary and sufficient for a set of complex numbers to be the zeros of a polynomial of the form ω c(λ).
Vyacheslav Pivovarchik

A Note on Indefinite Douglas’ Lemma

The Douglas lemma on majorization and factorization of Hilbert space operators is extended to the setting of Krein space operators.
Leiba Rodman

Some Basic Properties of Polynomials in a Linear Relation in Linear Spaces

The behavior of the domain, the range, the kernel and the multivalued part of a polynomial in a linear relation is analyzed, respectively.
Adrian Sandovici
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