  In 1987, Shapiro shew that composition operator induced by symbol $\phi$ is
compact on the Lipschltz space if and only if the infinity norm of $\phi$ is
less than 1 by a spectral-theoretic argument, where $\phi$ is a holomorphic
self-map of the unit disk. In this paper, we shall generalize Shapiro's result
to the $n$-dimensional case.
