Self-adjoint Analytic Operator Functions: Local Spectral Function and Inner Linearization

摘要

In this note we continue the study of spectral properties of a self_adjoint analytic operator function A(z) that was started in [5]. It is shown that if A(z) satisfies the Virozub-Matsaev condition on some interval _¤_0 and is boundedly invertible in the endpoints of _¤_0, then the 'embedding' of the original Hilbert space H into the Hilbert space F, where the linearization of A(z) acts, is in fact an isomorphism between a subspace H(_¤_0) of H and .F. As a consequence, properties of the local spectral function of A(z) on ,_0 and a so-called inner linearization of the operator function A(z) in the subspace H(_¤_0) are established.