  A Toeplitz operator $T_\varphi$, $\varphi \in L^\infty(\mathbb{T}^n)$, is a
partial isometry if and only if there exist inner functions $\varphi_1,
\varphi_2 \in H^\infty(\mathbb{D}^n)$ such that $\varphi_1$ and $\varphi_2$
depends on different variables and $\varphi = \bar{\varphi}_1 \varphi_2$. In
particular, for $n=1$, along with new proof, this recovers a classical theorem
of Brown and Douglas.
  \noindent We also prove that a partially isometric Toeplitz operator is
hyponormal if and only if the corresponding symbol is an inner function in
$H^\infty(\mathbb{D}^n)$. Moreover, partially isometric Toeplitz operators are
always power partial isometry (following Halmos and Wallen), and hence, up to
unitary equivalence, a partially isometric Toeplitz operator with symbol in
$L^\infty(\mathbb{T}^n)$, $n > 1$, is either a shift, or a co-shift, or a
direct sum of truncated shifts. Along the way, we prove that $T_\varphi$ is a
shift whenever $\varphi$ is inner in $H^\infty(\mathbb{D}^n)$.
