  We consider the bounded linear operators with domain in the Hilbert space
$L^2(S^n)$, $n=2,3,5$ and describe its symbolic calculus defined by the Berezin
quantization. In particular, we derive an explicit formula for the composition
of Berezin's symbols and thus a noncommutative invariant star product, which in
turn is invariant under the action of the group $SU(2)$, $SU(2)\times SU(2)$
and $SU(4)$ on $\mathbb C^2$ , $\mathbb C^4$ and $\mathbb C^8$ respectively.
