Spectral Components of Selfadjoint Block Operator Matrices with Unbounded Entries

Abstract

Abstract 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 H 1 × H 2 where the entries A, B and C are not necessarily bounded operators in the Hilbert spaces H 1 , H 2 or between them, respectively. Under suitable assumptions a selfadjoint operator L is associated with L o 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 σ( C ) ≤ α ≤ min σ(A), descriptions of the spectral subspaces of L corresponding to the intervals [— ∞, α] and ]α,∞[ 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.