For a class of selfadjoint operators in a Krein space containing the definitizable selfadjoint operators a funetional calculus and the spectral function are studied Stability properties of the spectral function with respect to small compact perturbations of the resolvent are proved

On a class of selfadjoint operators in Krein space and their compact perturbations

The aim of this paper is to prove two perturbation results for a selfadjoint operator A in a Krein space

 $$
\mathcal{H}
$$

 which can roughly be described as follows: (1) If Δ is an open subset of

 $$
{\ifmmode\expandafter\overline\else\expandafter\=\fi{\mathbb{R}}}
$$

 and all spectral subspaces for A corresponding to compact subsets of Δ have finite rank of negativity, the same is true for a selfadjoint operator B in

 $$
\mathcal{H}
$$

 for which the difference of the resolvents of A and B is compact. (2) The property that there exists some neighbourhood Δ∞ of ∞ such that the restriction of A to a spectral subspace for A corresponding to Δ∞ is a nonnegative operator in

 $$
\mathcal{H},
$$

 is preserved under relative

 $$
\mathfrak{G}_p
$$

 perturbations in form sense if the resulting operator is again selfadjoint. The assertion (1) is proved for selfadjoint relations A and B. (1) and (2) generalize some known results.

On Compact Perturbations of Locally Definitizable Selfadjoint Relations in Krein Spaces

Local Scattering Problem and a Solvable Model of Quantum Network- Quadratic (Weakly) Hyperbolic Matrix Polynomials: Direct and Inverse Spectral Problems- A Theorem on Existence of Invariant Subspaces for J-binoncontractive Operators- Moment Problems for Real Measures on the Unit Circle- Finite Rank Perturbations, Scattering Matrices and Inverse Problems- Conditions for an Indefinite Sturm-Liouville Riesz Basis Property- Convergence of Diagonal Pade Approximants for a Class of Definitizable Functions- Approximation of N ? ? -functions II: Convergence of Models- Nonnegative Extensions via Embeddings- Fredholm Properties of Unbounded Operators on Interpolation Spaces- Domain and Range Descriptions for Adjoint Relations, and Parallel Sums and Differences of Forms- The Konigs Problem and Extreme Fixed Points- Bisectors, Isometries and Connected Components in Hilbert Spaces- On Norms in Indefinite Inner Product Spaces- Interpolation of Sobolev Spaces and Indefinite Elliptic Spectral Problems- On the Spectrum of a Quantum Dot with Impurity in the Lobachevsky Plane- Note on Interlacing for Hyperbolic Quadratic Pencils

http://www.gbv.de/dms/ilmenau/toc/602554489.PDF

Recent Advances in Operator Theory in Hilbert and Krein Spaces

Lef 9(J€) denote the algebra of all bounded linear operators on a separa- ble Hilbert space .l€. The first objective of this note is to study the class Ds(9(.r€)) of all 9(J€)-va\rcd functions tr. meromorphic in the open disc D and holomorphic at 0, admitting a representation of the form

/pdf/a-class-of-operator-valued-meromorphic-functions-on-the-unit-azad0fbwnk.pdf

A class of operator.valued meromorphic functions on the unit disc

We investigate some classes of eigenvalue dependent boundary value problems of the form 






where A ⊂ A+ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ0, Γ1 are abstract boundary mappings. It is assumed that A admits a self-adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self-adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self-adjoint extension A of A × T in K × H with the property that the compressed resolvent PK (A – λ)–1|Kk yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so-called linearization A are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

/pdf/boundary-value-problems-with-eigenvalue-depending-boundary-1domgu0qng.pdf

Boundary value problems with eigenvalue depending boundary conditions

Let H be a Krein space and U a unitary operator in H. Assume that for a certain open subset ¦£ of the unit circle T no point of Γ is accumulation point of the nonunitary spectrum σ(U)\T of U and that every point of Γ can be connected with 0 and ∞ by curves in the resolvent set ρ(U). Such an operator is called definitizable over Γ, if, roughly speaking, it has a spectral function over ¦£ (in the sense of the theory of definitizable operators). This class of operators, of course, contains the definitizable unitary operators in H. The main concern of this paper is to study some aspects of the behaviour of the spectral functions under compact perturbations of the corresponding operators within this class.

On a Class of Unitary Operators in Krein Space

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On a Class of Unitary Operators in Krein Space