  In this paper, we investigate the spectra of invertible weighted composition
operators with automorphism symbols, on Hardy space $H^2(\mathbb{B}_N)$ and
weighted Bergman spaces $A_\alpha^2(\mathbb{B}_N)$, where $\mathbb{B}_N$ is the
unit ball of the $N$-dimensional complex space. By taking $N=1$,
$\mathbb{B}_N=\mathbb{D}$ the unit disc, we also complete the discussion about
the spectrum of a weighted composition operator when it is invertible on
$H^2(\mathbb{D})$ or $A_\alpha^2(\mathbb{D})$.
