We give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

中文翻译：

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具有无穷系数的高阶微分方程的正确性条件

我们给出了希尔伯特空间中具有无穷大系数的高阶线性微分方程解的存在性和唯一性的充分条件。我们获得了有关解决方案及其导数的加权范数的一些估计。使用这些估计，我们显示了与分解子相关的一些积分算子的紧致性的条件。