Hosein Ansari - Depatment of mathematics,Islamic Azad university,Ahar Branch,Ahar,Iran.
Siavash ghorbanian - Depatment of mathematics,Islamic Azad university, Firoozkooh Branch.Firoozkooh, Iran

In the present paper we consider the Dirichlet problem for the differential-operator equation of the fourth order where , is a linear opeator in the separable Hilbert space and has a complete system of the eigenfunctins, which form a Riesz basis in We prove the existence and uniqueness of the generalized solution for the Dirichlet problem and give the description of the spectrum for the corresponding operator.

Dirichlet problem, weighted Sobolev spaces, differential equations in abstract spaces, spectrum of the linear operator