Self-adjoint analytic operator functions: Local spectral function and inner linearization

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 Δ₀ and is boundedly invertible in the endpoints of Δ₀, 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 HΔ0 of H and F. As a consequence, properties of the local spectral function of A(z) on Δ₀ and a so-called inner linearization of the operator function A(z) in the subspace H Δ0are established.