Representations of the solutions for volterra integro-differential equations in hilbert spaces

Volterra integro-differential equations with operator coefficients in Hilbert spaces were studied. The relationship has been established between the spectra of operator functions that are the symbols of the specified integro-differential equations and the spectra of generators of semigroups. Representations of solutions for considered integro-differential equations are obtained on the basis of spectral analysis of generators of operator semigroups and corresponding operator-functions.