  The $\Pi$-operator (Ahlfors-Beurling transform) plays an important role in
solving the Beltrami equation. In this paper we define two $\Pi$-operators on
the n-sphere. The first spherical $\Pi$-operator is shown to be an $L^2$
isometry up to isomorphism. To improve this, with the help of the spectrum of
the spherical Dirac operator, the second spherical $\Pi$ operator is
constructed as an isometric $L^2$ operator over the sphere. Some analogous
properties for both $\Pi$-operators are also developed. We also study the
applications of both spherical $\Pi$-operators to the solution of the spherical
Beltrami equations.
