Self-adjoint analytic operator functions and their local spectral function

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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 ) = λ I − 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.