We dedicate this paper to the memory of MARK ALEXANDROVICH KRASNOSEL'SKII -an outstanding mathematician and man Abstract.In some weighted La vector space we study a symmetric semibounded operator ILb which is given by a 3 x 3 system of ordinary differential operators on an interval1 [O,ro] with a singularity at r = 0 (see (0.1)). This system can be considered as a "smooth" perturbation of a more specific physical model describing the oscillations of plasma in an equilibrium configuration in a cylindrical domain (see (1.12)). This perturbation is smooth in the sense that in the system under study in comparison with the physical model only the smooth parts of the coefficients are changed conserving all types of singularities. It is the goal of this paper to construct a suitable selfadjoint extension L of the symmetric operator ILL (and its closure LO) and to determine the essential spectrum of this extension. The essential spectrum consists of two bands (which may overlap) if we exclude the singularities by considering the system on an interval (r1,ro] with 0 < ri < ro.In the corresponding physical model these bands are called Alf%n spectrum and slow magnetosonic spectrum. It is shown that the singularity in 0 generates additional components of the essential spectrum which under specific conditions, as in the case of the phyaical model, "disappear" in the two bands known from the "regular" case (r1,ro) with rl 0. IntroductionIn this paper we consider selfadjoint differential operators defined by a 3 x 3 block operator matrix 1991 Mathematics Subject Classification. 47A10, 47B25, 76W05.Keywords and phmses. Selfadjoint operator matrices, systems of singular ordinary differential operators, essential spectrum, magnetohydrodynamics.Math. Nachr. 205 (1999) Here r, &, P2, 71, P and dik (4 k = 1,2) are sufficiently smooth functions on the interval [0, 7-01. It is a system of ordinary differential operators of mixed order on the interval (O,ro] with a singularity at r = 0. This differential operator is related to a simple physical model describing the oscillations of plasma in an equilibrium configuration in a cylindrical domain. More precisely, the so-called force operator in cylindrical coordinates has the form (0.1) after having applying the Fourier mode decomposition, but with more specific coefficients which are presented in formula (1.12). Actually, the whole force operator is an orthogonal sum of an infinite family of operators of type (0.1) parametrized by two integers k and rn.The case of the force operator without singularity modelling the so-called hard core problem of a toroidal plasma configuration leads to the study of the operator (0.1) in the space ( L~( ( T I , T o ) ,~~~) )~ where 0 < TI < TO. In this hard core situation the plasma is kept between the (perfectly conducting) walls of two concentric tori with the same major and minor axes. This case has been investigated in numerous papers, first by physicists (see [HL], [Go], [Gr], for example) and later from the mathematical point of view (see [Kl], [D...

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Abstract. In this note we study the spectral properties of a multiplication operator in the space Lp(X)" which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called "floating singularities" .

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We construct operators which factorize the transfer function associated with a nonself-adjoint 2 × 2 operator matrix whose diagonal entries can have overlapping spectra and whose off-diagonal entries are unbounded operators.

We dedicate this paper to the memory of Selim Grigor'evich Krein -an outstanding mathematician and manThe famous classical S. Krein problem of small motions and normal oscillations of a viscous fluid in a partially filled container is investigated by a new approach based on a recently developed theory of operator matrices with unbounded entries. The initial boundary value problem is reduced to a Cauchy problem Ý Ø · Ý ´Øµ Ý´¼µ Ý ¼ in some Hilbert space. The operator matrix is a maximal uniformly accretive operator which is selfadjoint in this space with respect to some indefinite metric. The theorem on strong solvability of the hydrodynamic problem is proved. Further, the spectrum of normal oscillations, basis properties of eigenfunctions and other questions are studied.

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