On Fredholm Solvability of the Dirichet Problem for Linear Differential Equations of Infinite Order

We propose a new approach to investigation of solvability of the Dirichlet problem for differential equations of infinite order. Namely, by using the embedding theorems obtained by the author in previous papers for the energy spaces, corresponding to operators of infinite order, the initial differential operator of infinite order is expressed as a sum of the main and subordinate operators of infinite order. The conditions, under which the above Dirichlet problems are soluble, are established by using the main term of the corresponding differential operator.