Let H be a separable Hilbert space and H1=L2
(0, ∞ ;H). The all functions are defined in range[0, ∞) , their
values belongs to space H, they are measurable in the meaning of Bochner and
provides the condition of
If the scalar product is defined in H1 by the formula
f(x), g(x) ∈ H1, H1 forms a separable Hilbert space.
In this study, in space H1, it is investigated that Green`s function
(resolvant) of the operator formed by the differential expression.
(-1)n y (2n) +Q(x)y,
|0 ≤ x ≤ ∞
And boundary conditions
Y(j) (0)-h jy
| j = 1,3,......,2n-1
Where Q(x) is a normal operator that has pure degree spectrum for every x ∈
[0, ∞) ) in H. Assumed that domain of Q(x) is independent from x and resolvent
set of Q(x) belongs to
(0 < ∈ ≤ π) of Complex plane λ. In addition assumed that
the operator function Q(x) satisfies the Titchmars-Levitan complex conditions.
h j are arbitrary complex numbers. The obtained result has been applied
to an example.