On expansive operators that are quasisimilar to the unilateral shift of finite multiplicity

An operator $T$ on a Hilbert space $\mathcal{H}$ is called expansive, if $\mathrm{\parallel }Tx\mathrm{\parallel }\ge \mathrm{\parallel }x\mathrm{\parallel }$ ($x\in \mathcal{H}$). Expansive operators $T$ quasisimilar to the unilateral shift $S_N$ of finite multiplicity $N$ are studied. It is proved that $I-T^{\ast }T$ is of trace class for such $T$. Also the lattice $\mathrm{L}\mathrm{a}\mathrm{t}T$ of invariant subspaces of an expansive operator $T$ quasisimilar to $S_N$ is studied. It is proved that $\dim \mathcal{M}\ominus T\mathcal{M}\le N$ for every $\mathcal{M}\in \mathrm{L}\mathrm{a}\mathrm{t}T$. It is shown that if $N\ge 2$, then there exist ${\mathcal{M}}_j\in \mathrm{L}\mathrm{a}\mathrm{t}T$ ($j=1,\dots ,N$) such that the restriction $T{\mathrm{\mid }}_{{\mathcal{M}}_j}$ of $T$ on ${\mathcal{M}}_j$ is similar to the unilateral shift $S$ of multiplicity $1$ for every $j=1,\dots ,N$, and $\mathcal{H}={\vee }_{j=1}^N{\mathcal{M}}_j$. For $N=1$, that is, for $T$ quasisimilar to $S$, there exist two spaces ${\mathcal{M}}_1$, ${\mathcal{M}}_2\in \mathrm{L}\mathrm{a}\mathrm{t}T$ such that $T{\mathrm{\mid }}_{{\mathcal{M}}_j}$ 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 $\mathrm{L}\mathrm{a}\mathrm{t}T$ and $\mathrm{L}\mathrm{a}\mathrm{t}S$.