  We prove that the Markov-Stieltjes transform is a bounded non compact Hankel
operator on Hardy space $H^p$ with Hilbert matrix with respect to the standard
Schauder basis of $H^p$ and a bounded non compact operator on Lebesgue space
$L^p[0,1]$ for $p\in(1,\infty)$ and obtain estimates for its norm in this
spaces. It is shown that the Markov-Stieltjes transform on $L^2(0,1)$ is
unitary equivalent to the Markov-Stieltjes transform on $H^2$. Inverse formulas
and operational properties for this transform are obtained.
