H. Langer, M. Langer, A. Markus and C. Tretter,
The Virozub–Matsaev condition and spectrum of definite type for self-adjoint operator functions,
Complex Anal. Oper. Theory 2 (2008), 99–134

Abstract:
We establish sufficient conditions for the so-called Virozub–Matsaev condition for twice continuously differentiable self-adjoint operator functions. This condition is closely related to the existence of a local spectral function and to the notion of positive type spectrum. Applications to self-adjoint operators in Krein spaces and to quadratic operator polynomials are given.

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