  Let $(A_S,S)$ be an Artin-Tits and $X$ a subset of $S$ ; denote by $A_X$ the
subgroup of $A_S$ generated by $X$. When $A_S$ is of spherical type, we prove
that the normalizer and the commensurator of $A_X$ in $A_S$ are equal and are
the product of $A_X$ by the quasi-centralizer of $A_X$ in $A_S$. Looking to the
associated monoids $A_S^+$ and $A_X^+$, we describe the quasi-centralizer of
$A_X^+$ in $A_S^+$ thanks to results in Coxeter groups. These two results
generalize earlier results of Paris. Finaly, we compare, in the spherical case,
the normalizer of a parabolic subgroup in the Artin-Tits group and in the
  Coxeter group.
