Parabolicity of second order differrential equations in hilbert space

We study conditions entailing that the differnetial equation u″+Bu′+Au=0 in a Hilbert space generates a corresponding equation in a larger extrapolation space which is parabolic. Subsequently we prove some relations between solutions of the original equation and the extrapolated equation. An example of applications to initial boundary value problems for partial differential equations is given. Research partially supported by Ministero della Pubblica Istruzione, Italy (Fondi 40%), Università di Bologna (Fondi 60%), and the GNAFA of CNR.