Branko Najman

Remarks on the perturbation of analytic matrix functions III

As in [N], [LN] the Newton diagram is used in order to get information about the first terms of the Puiseux expansions of the eigenvalues λ(ε) of the perturbed matrix pencilT(λ, ε)=A(λ)+B(λ, ε) in the neighbourhood of an unperturbed eigenvalue λ(∈) ofA(λ). In fact sufficient conditions are given which assure that the orders of these first terms correspond to the partial multiplicities of the eigenvalue λ0 ofA(λ).

Quasi-Uniformly Positive Operators in Krein Space

Definitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces. A sufficient condition for definitizability of a selfadjoint operator A with a nonempty resolvent set ρ(A) in a Krein space (H,[·❘·]) is the finiteness of the number of negative squares of the form [Ax❘y] (see [10, p. 11]).

NONSINGULARITY OF CRITICAL POINTS OF SOME DIFFERENTIAL AND DIFFERENCE OPERATORS

For two different examples of positive definitizable operators in Krein spaces we prove regularity of the critical points 0 and ∞. The first example is the Sturm-Liouville operator in L 2(ℝ,ρ) with ρ(x) = 007C;x007C; s sgn x. The second example is a difference operator in a Krein space of complex sequences.

Spectral theory of the Klein-Gordon equation in Krein spaces

In this paper the spectral properties of the abstract Klein–Gordon equation are studied. The main tool is an indefinite inner product known as the charge inner product. Under certain assumptions on the potential V , two operators are associated with the Klein–Gordon equation and studied in Krein spaces generated by the charge inner product. It is shown that the operators are self-adjoint and definitizable in these Krein spaces. As a consequence, they possess spectral functions with singularities, their essential spectra are real with a gap around 0 and their non-real spectra consist of finitely many eigenvalues of finite algebraic multiplicity which are symmetric to the real axis. One of these operators generates a strongly continuous group of unitary operators in the Krein space; the other one gives rise to two bounded semi-groups. Finally, the results are applied to the Klein–Gordon equation in Rn.

Perturbation of the eigenvalues of quadratic matrix polynomials

Perturbation properties of a quadratic matrix pencil containing a “small” parameter are considered. Main results concern the splitting properties of multiple eigenvalues and the corresponding Puiseux expansions. For hermitian pencils it is proved that the Puiseux expansion generates groups of eigenvalues ordered according to the partial algebraic multiplicities of the unperturbed eigenvalue.

Positive Differential Operators in Krein Space L2(ℝ)

Consider the weighted eigenvalue problem

LEADING COEFFICIENTS OF THE EIGENVALUES OF PERTURBED ANALYTIC MATRIX FUNCTIONS

Lu = {\rm \lambda }\left( {{\rm sgn}\;{\rm x}} \right)u,

Some Interlacing Results for Indefinite Hermitian Matrices

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