Oscillation of Operator Differential Equations in Hilbert Space

where P: R+→L(H,H) is continuous and self-adjoint for each t∈R+=[0,∞). By a solution of (I), we mean any function X: R+→L(H,H) which satisfies (I) on R+. Here X" denotes the second strong derivative of X. Any solution X(t) of (I) satisfies X*(t)X'(t)=X*'(t)X(t), t∈R+ is called to be prepared. A prepared solution of (I) is nonoscillatory if there exists t0≧0 such that, for every t≧t0 X(t) is an isomorphism, i.e., X(t) is one-to-one and onto. Otherwise X(t) is said to be oscillatory. Equation (I) is oscillatory if every prepared solution of (I) is oscillatory. A bounded linear functional f: L(H,H)→R is said to be positive if f(T)≧0 for every positive operator T∈L(H,H), i.e., any T with ≧0 for all x∈H, and f(A*B)=f(B*A) for every A, B∈L(H,H). Here <,> denotes the inner product of H. Every positive linear functional f satisfies ‖f‖=f(I) and if f is not the zero functional, it satisfies