Let ℋ be a Hilbert space
and let ℬ = ℬ(ℋ,ℋ) be the Banach algebra of bounded linear operators
from ℋ to ℋ with the usual operator norm. Let 𝒮 be the subspace of ℬ
consisting of the self adjoint operators, and consider the second order differential
system
(1)
on R+= [0,∞), where P : R+→𝒮 is continuous. Let 𝒢 be the set of positive linear
functionals on ℬ. The elements of 𝒢 are used to derive oscillation criteria for this
differential system. These criteria include a large number of well-known oscillation
criteria for corresponding matrix differential systems and scalar equations. Extensions
of the results to nonlinear differential systems and differential inequalities are also
discussed.
