  For $\mu>\beta>0$, the generalized Stieltjes operators $$
\mathcal{S}_{\beta,\mu} f(t):={t^{\mu-\beta}}\int_0^\infty {s^{\beta-1}\over
(s+t)^{\mu}}f(s)ds, \qquad t>0, $$
  defined on Sobolev spaces $\mathcal{T}_p^{(\alpha)}(t^\alpha)$ (where
$\alpha\ge 0$ is the fractional order of derivation and these spaces are
  embedded in $L^p(\RR^+)$ for $p\ge 1$) are studied in detail. If $0 < \beta -
\pp < \mu$, then operators $\mathcal{S}_{\beta,\mu}$ are
  bounded (and we compute their operator norms which depend on $p$); commute
and factorize with generalized Ces\'{a}ro operator on
$\mathcal{T}_p^{(\alpha)}(t^\alpha)$ . We calculate and represent explicitly
  their spectrum set $\sigma (\mathcal{S}_{\beta,\mu})$. The main technique is
to subordinate these operators in terms of $C_0$-groups and transfer new
properties from some special functions to Stieltjes operators. We also prove
some similar results for generalized Stieltjes operators $
\mathcal{S}_{\beta,\mu}$ in the Sobolev-Lebesgue
$\mathcal{T}_p^{(\alpha)}(\vert t\vert^\alpha)$ defined on the real line $\R$.
We show connections with the Fourier and the Hilbert transform and a
convolution product defined by the Hilbert transform.
