Self-adjoint analytic operator functions and their local spectral function

For a self-adjoint analytic operator function A(λ), which satisfies on some interval Δ of the real axis the Virozub–Matsaev condition, a local spectral function Q on Δ, the values of which are non-negative operators, is introduced and studied. In the particular case that A(λ)=λI−A with a self-adjoint operator A, it coincides with the orthogonal spectral function of A. An essential tool is a linearization of A(λ) by means of a self-adjoint operator in some Krein space and the local spectral function of this linearization. The main results of the paper concern properties of the range of Q(Δ) and the description of a natural complement of this range.