Green functions of sturm-liouville equation with operator coffecient and seperation problem

SUMMARY Let H be seperable Hubert space. Let us represent the set of functions by Hi = L2(a, b; H) which is defined in the interval a < x < 6(- oo < a < x < b < oo) whose values belong to H, strongly measurable and satisfying the condition J a \\f{x)\\]idx < 00 E we define the inner product of f(x) and g(x) functions belonging to Hi, by the formula (f,9h = / (f{x),g{x))Hdx Ja Hi forms a seperable Hubert space. In the first chapter, Green function of L operator formed by -y" + Q(x)y Strum-Liouville differantial expression in Hi = £2(0,00; H) space and when h is complex number, y'(0) - hy(0) = 0 boundary condition is studied. In the second chapter, seperation theorem for the operator formed by -y" + Q(x)y expression in £r2(- 00, 00; H) space has been proved. Also in both chapter, Q(x) is an operator which transforms at H in each value of a;, is self adjoint, lower bounded and its inverse is completely continous. Ill