Let H be a complex separable Hilbert space and let L(H) be the algebra of all bounded linear operators on H. The boundary value problem (1) \(X^{(n)}+A_{n-1}X^{(n-1)}+...+A_ 0X=0\), (2) \(EX(b)-X(0)F=G\), where \(A_ i,E,F,G\in L(H)\) and b is a positive real number is considered. Explicit expressions for solutions (in L(H)) of the BVP (1), (2) and the Cauchy problem for equation (1) are given in terms of solutions (in L(H)) of the algebraic equation (3) \(X^ n+A_{n- 1}X^{n-1}+...+A_ 0X=0.\) A method for solving equation (3) by reducing the degree of the equation is studied. This reduction is based on the application of annihilating analytic functions of operators. The paper can be regarded as a continuation of the recently author's paper ``Explicit solutions for second order operator differential equations with two boundary value conditions'' [Linear Algebra Appl. 103, 73-86 (1988)].