This paper is concerned with certain estimates on the asymptotic behaviour of the functions [italic]u defined on an interval (a, [infinity symbol]) with values in a Hilbert space [italic]H. More precisely, if [italic]L is a second order ordinary differential operator the coefficients of which are operators acting in [italic]H, we wish to obtain inequalities allowing one to get information about the behaviour of a function [italic]u in a neighborhood of infinity from the asymptotic behaviour of the function [italic]L[italic]u. These inequalities will be called Hardy type inequalities.