Eigenfunction Expansions Associated with Second-order Differential Equations for Hilbert Space-valued Functions

where for each r € HO, oo) A(r) is an operator in a Hilbert space H and & acts on ^-valued functions on fO, co). Restricting the domain of <£ appropriately, we can regard 3? as an operator in f) = £2 (0, °° ; H}. Our purpose is to develop an eigenfunction expansion theory associated with the differential operator <£. If dim H=l, i.e. jfiT=C, then & is an ordinary second-order differential operator and A(r) is simply an operator of multiplication by a function g(r). For real-valued g(r) a rather complete eigenfunction expansion theory has been worked out by Weyl ^10], Stone Q8j, Titchmarsh Q9j, Kodaira £4], Q5] and others. But when H is an infinite-dimensional Hilbert space, it seems that no complete theory, comparable with the one for ordinary differential operators, has been presented. Rofe-Beketov Q7] considers the case where A(f) is a bounded selfadjoint operator-valued function on QO, oo) which is continuous in the uniform operator topology. He shows that there exist a non-negative definite, bounded opera tor -valued, interval function p(/), /CR? and a bounded operator-valued function a)(r, ^) on Q03 oo)5 satisfying