Let ℋ be a Hilbert space and
let ℬ = ℬ(ℋ,ℋ) be the B∗-algebra of bounded linear operators from ℋ to ℋ with the
uniform operator topology. Let 𝒮 be the subset of ℬ consisting of the selfadjoint
operators. This paper is concemed with second order, selfadjoint differential
equations of the form
(1)
on ℛ+= [0,∞), where P and Q are continuous mappings of ℛ+ into 𝒮 with P(x)
positive definite for all x ∈ℛ+. Let 𝒢 be the set of positive linear functionals on ℬ.
Positive functionals are used in deriving a generalization of Sturm’s comparison
theorem, and, in turn, the comparison theorem is used to obtain oscillation criteria
for equation (1). These criteria are shown to include a large number of well-known
oscillation criteria for (1) in the matrix and scalar case. Extensions of the
results to nonlinear differential equations and differential inequalities are also
discussed.