Explicit solutions for second order operator differential equations with two boundary value conditions (Q1106388)

Let H be a separable, complex Hilbert space and L(H) the algebra of all bounded linear operaors on H with the operator norm. In an analogoues way to the scalar case different fundamental sets of solutions for an equation \(X''+A_ 1X'+A_ 0X=0\) \((A_ 0,A_ 1\in L(H)\), ' denote the strong derivative) are obtained in terms of solutions of the corresponding algebraic equation \(X^ 2+A_ 1X+A_ 0=0\). The main result is an explicit expression for the solution of the nonhomogeneous boundary value problem \(X''+A_ 1X'+A_ 0X=F(t),\) \(X(b)-X(0)=H_ 0,\) \(X'(b)-X'(0)=H_ 1,\) \((b>0)\) \((A_ 0,A_ 1,H_ 0,H_ 1\in L(H),F: [0,b]\to L(H)\) continuous operator function) by a Green operator function. The transform of results from the scalar case is not trivial. This paper may be regarded as a continuation of a former author's paper [ibid. 83, 29-38 (1986; Zbl 0614.34030)].