Reinhard Mennicken

Boundary value problems for a class of elliptic operator pencils.

In this paper operator pencilsA(x, D, λ) are studied which act on a manifold with boundary and satisfy the condition of N-ellipticity with parameter, a generalization of the notion of ellipticity with parameter as introduced by Agmon and Agranovich-Vishik. Sobolev spaces corresponding to the Newton polygon are defined and investigated; in particular it is possible to describe their trace spaces. With respect to these spaces, an a priori estimate is proved for the Dirichlet boundary value problem connected with an N-elliptic pencil.

On the Spectrum of the Product of Closed Operators

In this note we study the connection between the spectra of the products AB and BA of unbounded closed operators A and B acting in Banach spaces. Under the condition that the resolvent sets of these products are not empty we show that the spectra of AB and BA coincide away from zero and prove the commutation relation . Further, we prove statements concerning the relationship between the spectra of the operator AB and the block operator matrix .

On elliptic operator pencils with general boundary conditions

In this paper parameter-dependent partial differential operators are investigated which satisfy the condition of N-ellipticity with parameter, an ellipticity condition formulated with the use of the Newton polygon. For boundary value problems with general boundary operators we define N-ellipticity including an analogue of the Shapiro-Lopatinskii condition. It is show that the boundary value problem is N-elliptic if and only if an a priori estimate with respect to certain parameter-dependent norms holds. These results are closely connected with singular perturbation theory and lead to uniform estimates, for problems of Vishik-Lyusternik type containing a small parameter.

On the Friedrichs extension of some block operator matrices

We give a matrix representation for the resolvent of the Friedrichs extension of some semibounded 2×2 operator matrices and study their essential spectrum.

The Essential Spectrum of a System of Singular Ordinary Differential Operators of Mixed Order. Part II: The Generalization of Kako's Problem

A system of ordinary differential equations of mixed order on an interval (0, r0) is considered, where some coefficients are singular at 0. Special cases have been dealt with by Kako, where the essential spectrum of an operator associated with a linearized MHD model was calculated, and more recently by Hardt, Mennicken and Naboko. In both papers this operator is a selfadjoint extension of an operator on sufficiently smooth functions. The approach in the present paper is different in that a suitable operator associated with the given system of ordinary differential equations is explicitly defined as the closure of an operator defined on sufficiently smooth functions. This closed operator can be written as a sum of a selfadjoint operator and a bounded operator. It is shown that its essential spectrum is a nonempty compact subset of ℂ, and formulas for the calculation of the essential spectrum in terms of the coefficients are given.

Essential spectrum of a system of singular differential operators and the asymptotic Hain-Lust operator

We consider a matrix differential operator with singular entries which arises in magnetohydrodynamics. By means of the asymptotic HainLust operator and some pseudo-differential operator techniques, we determine the essential spectrum of this operator. Whereas in the regular case, the essential spectrum consists of two intervals, it turns out that in the singular case two additional intervals due to the singularity may arise. In addition, we establish criteria for the essential spectrum to lie in the left half-plane.

On the Spectrum of Unbounded Off-diagonal 2 × 2 Operator Matrices in Banach Spaces

In the product of two Banach spaces we study the spectrum of nonselfadjoint 2 × 2 off-diagonal operator matrices

On the essential spectrum of the linearized Navier-Stokes operator

We determine the essential spectrum of the linearized Navier-Stokes operator with physical boundary conditions. In contrast to other approaches we do not make use of pseudo-differential operators. We establish a direct proof using only some fundamental results for matrix operators.

On the Essential Spectrum of a Differentially Rotating Star

Natural oscillations of a differentially rotating star are governed by the linearized Euler equations. Separation of variables leads to a family L k,0 (k ∈ Z) of mixed order partial differential operators. It is shown that for k ¬= 0 their closures L k have nonempty essential spectrum. Indeed, it is shown that the essential spectrum of L k coincides with the essential spectrum of a bounded operator. Some parts of the essential spectrum are calculated explicitly. It is still an open problem if there are more points in the essential spectrum.

Contributions to operator theory in spaces with an indefinite metric

Heinz Langer and his work, A. Dijksma, I. Gohberg on the spectra of some class of quadratic operator pencils, V. Adamyan, V. Pivovarchik special realizations for Schur upper triangular operators, D. Alpay, Y. Peretz on the defect of noncontractive operators in Krein spaces -a new formula and some applications, T. Ya et al positive differential operators in the Krein space L2(Rn), B. Curgus, B. Najman singular values of positive pencils and applications, R.L. Ellis et al perturbations of Krein spaces preserving the nonsingularity of the critical point infinity, A. Fleige, B. Najman an analysis of the block structure of jqq-inner functions, B. Fritzsche et al selfadjoint extensions of the orthogonal sum of symmetric relations II, S. Hassi et al some interpolation problems of Nevanlinna-Pick type - the Krein-Langer method, s. Hassi et al on the spectral representation for singular selfadjoint boundary eigenvalue problems, D. Hinton, A. Schneider some characteristics of a linear manifold in a Krein space and their applications, E.I. Iokvidow riggings and relatively from bounded perturbations of nonnegative operators in Krein spaces, P. Jonas norm bounds for Volterra integral operators and time-varying linear systems with finite horizon, M.A. Kaashoek, A.C.M. Ran the numerical range of selfadjoint matrix polynomials, P. Lancaster et al spectral properties of a matrix polynomial connected with a component of its numerical range, A. Markus et al Lyapunov stability of a multiplication operator perturbed by a Volterra operator, S. Naboko, C. Tretter Multiplicative perturbations of positive operators in Krein spaces, B. Najman, K. Veselic on the number of negative squares of certain functions, Z. Sasvari factorization of elliptic pencils and the Mandelstam hypothesis, A.A. Shkalikov an inductive limit procedure within the quantum harmonic oscillator, F.H. Szafraniec canonical systems with a semibounded spectrum, H. Winkler.