of Vienna, R. MENNICKEN') of Regensburg and A. A. SHKALIKOV3) of Moscow Dedicated to Professor ISRAEL GOHBERG on the occassion of his 65th birthday (Received July 23, 1993) IntroductionIn this note we consider operators Lo defined by a 2 x 2 block operator matrix where the entries are in general unbounded operators, A acting in a Banach space X , , D acting in a Banach space X , , and B, C acting between these spaces. Apart from other assumptions formulated below we always assume that Then the matrix in (0.1) defines a linear operator in X = X , x X , with domain 9 ( A ) x 9 ( B ) . Operators of this form arise in magnetohydrodynamics, astrophysics and fluid mechanics, see e.g.[A], [GI, [L]. In these applications A, B and C are differential operators with B, C of lower order than A, and D is a differential operator or a multiplication operator.In general, the operator Lo as defined in (0.1) is not closed or closable, even if its entries are closed. Therefore it is of interest to find conditions under which Lo is closable and to describe its closure, which we shall denote by L. The second question we are concerned with in this paper is to determine the essential spectrum of the operator L. If A is an operator with compact resolvent (and hence with a discrete spectrum) and D has a nonempty essential spectrum, then the essential spectrum of L is in general nonempty. However, if B or C is unbounded, it does not necessarily coincide with the essential spectrum of D.') F. V. ATKINSON gratefully acknowledges an ALEXANDER-VON-HUMBOLDT Research

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The authors study symmetric operator matrices A B = ( B ' C ) in the product of Hilbert spaces H = Hi xH2, where the entries are not necessarily bounded operators. Under suitable assumptions the closure Lo exists and is a selfadjoint operator in H. With Lo, the closure of the transfer function M(X) = C -X -B * ( A -X)-'B is considered. Under the assumption that there exists a real number p < inf p(A) such that M ( P ) << 0, it follows that E p ( z). Applying a factorization result of A. I. VIROZUB and V. I. MATSAEV [VM] to the holomorphic operator function M( A), thespectral subspaces of corresponding to the intervals ] -00, p ] and [ p, 00[ and the restrictions of Lo to these subspaces are characterized. Similar results are proved for operator matrices which are symmetric in a Kr&n space. 0. Introduction In this note we consider 2 x 2 operator matrices of the form and A B To = ( -B * c ) 1991 Mathematics Subject ClassIficatIon. Primary 47B 25; Secondary 47A 11, 47 A56, 47B 50, Keywords and phrases. Selfadjoint operator matrices, transfer functions, half range completeness, 35 P 10. eigenfunction expansions for PDO. (1973), 79 -93 61, NO 2 (1988), 289-307 716 -746

We consider operator matrices H = ( Blo '01 Al ) with self-adjoint entries A ; , i = 0,1, ead bounded Bol = Bio, acting in the orthogonal rum 31 = 36 @ 311 of Hilbert spacea % and 311. We are especially interested in the eyc where the rpsctrum of, w, A1 ir partly or totally embedded into the continuous spectrum of A0 m d the t r d e r fuoction M~( z )admits mdytic continuation (M an o p e r a t o r -d u d function) through the cuts dong branchea of the continuous spectrum of the entry & into the unphysical wheet(8) of the spectral parameter plane. The d u a s of L in the unphylicrl beet# whom MT1(z) and coneequently the m l v e n t (H -I)-' have polas are w u d y u l l d raonums. A main god of the present work is to 6nd non-selfadjoint operators whore rpactra include the re8onanm an 4 IB to study the completeneea and basis properties of the rasonmce eigenvactars of M l ( z ) in 311. 'It, this end we first construct an operator-valued function R ( Y ) on the #pace of operators in 311 possessing the property: Vl(Y)+l = V~(Z)$I for m y eigenvector $1 of Y corrasponding to an eigendue z and then study the quation H1 = A1 + R(H1). We prove the d n b i l i t y of this quation even in the c a~e where the spectra of A0 and A1 overlap. Uiing the frct that the root vectors of the wlutiom H1 are at the uune time such vectors for M1 (z), we prove completeneon and even b i s propertiea for the root vectors (including those for the resonmces). 1991 M a h m o t i c r Subject Clorsifiation. Primary: 47A56,47Nxx; Secondary: 47N5OI47A40. for any eigenvector $('I corresponding to an eigenvalue z of the operator K. The desired operator Hi was searched for as a solution of the operator equation ( 1-81 Hi = A, + K ( H i ) , i = 0, 1. Notice that an equation of the form (1.8) first appeared explicitly in the paper [7] by M. A. BRAUN. Obviously, if Hi is a solution of Eq. (1.8) and Hi$(i) = z$J(') then, due to (1.7), automatically a$(') = (A< + &(Hi))$(i) = (Ai + &(z))$(') and, thus, for these z and $('I the equality (1.5) holds. The solvability of the equation (1.8) was announced in [29] and proved in [28, 301 under the assumption wketfi %r the Hifbert-Schmidt norm of the couplings Bi,. It was found (28, 301 that the problem of constructing the operators Hi is closely related to the problem of searching for the invariant subspaces 9il i = 0,1, of the matrix H which admit the graph representations A = { u E H : u = ( * , U , O , , ) , U o E 3 1 0 } 9 91 = ( u E 3 1 : u = ( Q O :~) , u 1 € H 1 } , (1.10) strictly below the spectrum of the other one, say (1.15) maxo(A1) < mina(A0). Soon, the result of [2] was extended by V. M. ADAMYAN, H. LANGER, R. MENNICKEN and J . SAURER [3] to the case where (1.16) maxa(A1) 5 mino(A0) and where the couplings Bij were allowed to be unbounded operators such that, for < mino(&), the product (& -a~)-'/~Bol makes sense as a bounded operator. The conditions (1.15), (1.16) were then somewhat weakened by R. MENNICKEN and A. A. SHKALIKOV [27] in the case of a bounded entry A1 and the same type of entries Bij a...

This paper is devoted to the study of the spectral components of selfadjoint operator matrices which are generated by symmetric operator matrices of the form in the product Hilbert space 'HI x ? i 2 where the entries A , B and C are not necessarily bounded operators in the Hilbert spaces 'HI, 312 or between them, respectively. Under suitable assumptions a selfadjoint operator L is associated with Lo and the spectral properties of L are studied. The main result concerns the case in which the spectra of the selfadjoint operators A and C are weakly separated. If a is a real number such that max u ( C ) 5 a 5 min u(A), descriptions of the spectral subspaces of L corresponding to the intervals ] -00, a ] and ]a, m[ and of the restrictions of L to these subspaces are given. From this main result half range completeness and basis properties for certain parts of the spectrum of L are deduced. The paper closes with two applications to systems of differential operators from magnetohydrodynamics. 0.In (0.1) 2 representations of the compressed resolvents of L are given which correspond to the first and second components in (0.1). Formally these compressed resolvents are the inverses of the Schur complements, however, the unboundedness of the operators A , B , C makes the formulas more involved. With these compressed resolvents also the corresponding compressed spectral functions of L are introduced. Starting from Section 3, the assumption that the spectra of A and C are separated is imposed. If a is as above, the kernel of La is described. These results are needed in Section 4 where by means of the Stieltjes inversion formula and the representations of the compressed resolvents of Section 2 the basic estimates of the compressed spectral functions are proved (Theorems 4.2 and 4.3). In Section 5 these estimates yield the angular operator representation of the spectral subspaces C -:= 4m,(l~ and C+ := C],,,[. This representation allows a description of the restrictions Llc-, LJL, as a perturbation of C in 3-12 and a perturbation of A in 711, respectively. In Sections 6 and 7 the spectral functions of these restrictions and the above mentioned half range completeness and basis properties are considered. Two applications to systems of ordinary and partial differential operators are considered in Sections 8 and 9.

Abstract. In the study of the resolvent of a scalar elliptic operator, say, on a manifold without boundary there is a well -known Agmon -Agranovich -Vishik condition of ellipticity with parameter which guarantees the existence of a ray of minimal growth of the resolvent. The paper is devoted to the investigation of the same problem in the case of systems which are elliptic in the sense of Douglis -Nirenberg. We look for algebraic conditions on the symbol providing the existence of the resolvent set containing a ray on the complex plane. We approach the problem using the Newton polyhedron method. The idea of the method is to study simultaneously all the quasihomogeneous parts of the system obtained by assigning to the spectral parameter various weights, defined by the corresponding Newton polygon. On this way several equivalent necessary and sufficient conditions on the symbol of the system guaranteeing the existence and sharp estimates for the resolvent are found. One of the equivalent conditions can be formulated in the following form: all the upper left minors of the symbol satisfy ellipticity conditions. This subclass of systems elliptic in the sense of Douglis -Nirenberg was introduced by A. Kozhevnikov [K2].

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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