SPECTRAL DECOMPOSITION OF SYMMETRICAL OPERATOR MATRICES

The authors study symmetric operator matrices L(o) = ((A)(B) (B)(C)) in the product of Hilbert spaces H = H-1 x H-2, where the entries are not necessarily bounded operators. Under suitable assumptions the clos ure (L(o)) over bar exists and is a selfadjoint operator in H. With (L (o)) over bar, the closure of the transfer function M(lambda) = C - la mbda -B(A - lambda)B--1 is considered. Under the assumption that ther e exists a real number beta < inf rho(A) such that M(beta) much less t han 0, it follows that beta is an element of rho((L(o)) over bar). App lying a factorization result of A.I. VIROZUB and V.I. MATSAEV [VM] to the holomorphic operator function M(lambda), the spectral subspaces of (L(o)) over bar corresponding to the intervals] - infinity, beta] and [beta, infinity[ and the restrictions of (L(o)) over bar to these sub spaces are characterized. Similar results are proved for operator matr ices T-o = ((A)(-B) (B)(C)) which are symmetric in a Krein space.