Differential equations with operator coefficients in Hilbert spaces (Q1326064)

This article deals with the equation \(\sum_{|\alpha|\leq m} A_ \alpha D^ \alpha u= f(x)\) \((x\in \mathbb{R}^ n)\) with bounded operators \(A_ \alpha\) from \(\mathbb{H}_{m-| \alpha|}\) into \(\mathbb{H}_ 0\), where \(\mathbb{H}_ m\subseteq\cdots\subseteq \mathbb{H}_ 1\subseteq \mathbb{H}_ 0\) are Hilbert spaces. The results of the article are some existence theorems in terms of the operator function \(R(\lambda)= \left[\sum_{|\alpha|\leq m}(i\lambda)^ \alpha A_ \alpha\right]^{-1}\).