ESSENTIAL NORM ESTIMATE OF A COMPOSITION OPERATOR BETWEEN BLOCH-TYPE SPACES IN THE UNIT BALL

Let Bn be the unit ball of Cn and φ = (φ1, . . . , φn) a holomorphic self-map of Bn. Let p, q > 0, be an estimate of the essential norm of a bounded composition operator Cφ induced by φ between the p-Bloch space βp(Bn) and q-Bloch space βq(Bn) given in this paper, as well as the corresponding results between the little p-Bloch space $\beta _0^p\left( {{B_n}} \right)$. As a consequence, a necessary and sufficient condition for the composition operator Cφ to be compact from βp(Bn) (or $\beta _0^p\left( {{B_n}} \right)$) into βq(Bn) (or $\beta _0^p\left( {{B_n}} \right)$) is obtained.