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

Abstract.

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 ${\mathcal{H}}$ into the Hilbert space ${\mathcal{F}}$, where the linearization of A(z) acts, is in fact an isomorphism between a subspace ${\mathcal{H}}(\Delta_{0})$ of ${\mathcal{H}}$ and ${\mathcal{F}}$. 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 ${\mathcal{H}}(\Delta_{0})$ are established.