  This paper proves two theorems. The first of these simplifies and lends
clarity to the previous characterizations of the invariant subspaces of $S$,
the operator of multiplication by the coordinate function $z$, on
$L^2(\mathbb{T};\mathbb{C}^n)$, where $\mathbb{T}$ is the unit circle, by
characterizing the invariant subspaces of $S^n$ on scalar valued $L^p$
($0<p\le\infty$) thereby eliminating range functions and partial isometries. It
also gives precise conditions as to when the operator shall be a pure shift and
describes the precise nature of the wandering vectors and the doubly invariant
subspaces. The second theorem describes the contractively contained Hilbert
spaces in $L^p$ that are simply invariant under $S^n$ thereby generalizing the
first theorem.
