Spectral analysis of integrodifferential equations in a Hilbert space

We investigate the correct solvability of initial-value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space. We do spectral analysis of operator-functions describing symbols of such equations. These equations are an abstract form of linear integrodifferential partial derivative equations arising in the viscoelasticity theory and having some other important applications. We establish the localization and the spectrum structure of operator-functions describing symbols of these equations.