The minimal index of a self-adjoint pencil

Let A and B be selfadjoint operators ona Hilbert space H. We define minimal index ν(A,B) = min {#negative eigenvalues of A - λB}, we connect it with various ideas in the literature and we connect it with formulae used in some recent variational principles

The minimal index of a self-adjoint pencil

Let A and B be selfadjoint operators ona Hilbert space H. We define minimal index ν(A,B) = min {#negative eigenvalues of A - λB}, we connect it with various ideas in the literature and we connect it with formulae used in some recent variational principles

Preservation of the Range Under Perturbations of an Operator

A sufficient condition for the stability of the range of a positive operator in a Hilbert space is given. As a consequence, we get a class of additive perturbations which preserve regularity of the critical point 0 of a positive operator in a Krein space

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

The Operator (sgn x) d²/dx² is Similar to a Selfadjoint Operator in L² (R)

Krein space operator-theoretic methods are used to prove that the operator (sgn x) d²/dx² is similar to a selfadjoint operator in the Hilbert space L²(R)

Convergence estimate for second order Cauchy problems with a small parameter

summary:We consider the second order initial value problem in a Hilbert space, which is a singular perturbation of a first order initial value problem. The difference of the solution and its singular limit is estimated in terms of the small parameter $\varepsilon .$ The coefficients are commuting self-adjoint operators and the estimates hold also for the semilinear problem