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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