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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 Berezinquantization. In particular, we derive an explicit formula for the compositionof Berezin's symbols and thus a noncommutative invariant star product, which inturn 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.

Star product induced from coherent states on $L^2(S^n)$, $n = 2, 3, 5$