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.
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