  An operator $T$ on a Hilbert space $\mathcal H$ is called expansive, if
$\|Tx\|\geq \|x\|$ ($x\in\mathcal H$). Expansive operators $T$ quasisimilar to
the unilateral shift $S_N$ of finite multiplicity $N$ are studied. It is proved
that $I-T^*T$ is of trace class for such $T$. Also the lattice $\mathrm{Lat}T$
of invariant subspaces of an expansive operator $T$ quasisimilar to $S_N$ is
studied. It is proved that $\dim\mathcal M\ominus T\mathcal M\leq N$ for every
$\mathcal M\in\mathrm{Lat}T$. It is shown that if $N\geq 2$, then there exist
$\mathcal M_j\in\mathrm{Lat}T$ ($j=1,\ldots, N$) such that the restriction
$T|_{\mathcal M_j}$ of $T$ on $\mathcal M_j$ is similar to the unilateral shift
$S$ of multiplicity $1$ for every $j=1,\ldots, N$, and $\mathcal
H=\vee_{j=1}^N\mathcal M_j$. For $N=1$, that is, for $T$ quasisimilar to $S$,
there exist two spaces $\mathcal M_1$, $\mathcal M_2\in\mathrm{Lat}T$ such that
$T|_{\mathcal M_j}$ is similar to $S$ for $j=1,2$, and $\mathcal H=\mathcal
M_1\vee\mathcal M_2$. Example of an expansive operator $T$ quasisimilar to $S$
is given such that intertwining transformations do not give an isomorphism of
$\mathrm{Lat}T$ and $\mathrm{Lat}S$.
