  The Friedrichs extension for the generalized spiked harmonic oscillator given
by the singular differential operator -D^2+ Bx^2 + Ax^{-2} + lambda x^{-alpha}
(B>0, A >= 0) in L_2(0, infinity) is studied. We look at two different domains
of definition for each of these differential operators in L_2(0, infinity),
namely C_0^infinity(0, infinity) and D(T_{2,F})\cap D(M_{lambda, alpha}), where
the latter is a subspace of the Sobolev space W_{2,2}(0, infinity). Adjoints of
these differential operators on C_0^infinity(0,infinity) exist as result of the
null-space properties of functionals. For the other domain, convolutions and
Jensen and Minkowski integral inequalities, density of
C_0^\infinity(0,\infinity) in D(T_{2,F})\cap D(M_{\lambda, \alpha}) in
L_2(0,\infinity) lead to the other adjoints. Further density properties
C_0^infinity(0,infinity) on D(T_{2,F})\cap D(M_{\lambda, \alpha}) yield the
Friedrichs extension of these differential operators with domains of definition
D(T_{2,F})\cap D(M_{lambda, alpha}).
