Degenerate differential-operator equations on infinite interval

In the present paper we consider the Dirichlet problem for the fourth order differential-operator equation $Lu\equiv(t^{\alpha}u^{\prime\prime})^{\prime\prime}+t^{-2}Au=f,$ where $t\in(1,~ +\infty),~\alpha\geq 2,~f\in L_{2,2}((1,~ +\infty),H),$ $A$ is a linear operator in the separable Hilbert space $H$ and has a complete system of eigenvectors that form a Riesz basis in $H.$ The existence and uniqueness of the generalized solution for the Dirichlet problem are proved, and the description of spectrum for the corresponding operator is given.