Properties of one higher order matrix-differential operator

The article considers a linear matrix-differential operator of the $n$-th order of the form $\mathbb{A}^n.$ For it and for the operator $(\tilde{\mathbb{A}}^{-1})^n,$ an analytical expression is derived, for which an operator analog of the Newton binomial is obtained. A lemma on the solution of a linear equation is given. It is used in the study of the abstract Cauchy problem for an algebro-differential equation in a Banach space with the cube of the operator $A$ at the highest derivative. The operator $A$ has the property of having $0$ as a normal eigenvalue. Conditions for the existence and uniqueness of the solution are determined; the solution is found, for which the method of cascade splitting of the equation and conditions into the corresponding equations and conditions in subspaces of lower dimensions is used. As an application, the results obtained for $n=3$ are used in solving a mixed problem for a fourth-order partial differential equation. These equations include the generalized shallow water wave equation and the generalized Liouville equation.