Quasisimilar weak contractions have isomorphic lattices of invariant subspaces

A contruction $T$ acting on a Hilbert space $H$ is called a weak contraction if the spectrum of $T$ does not cover the unit disk $\mathbb D$ and the operator $I-T^*T$ is of trace class. The operators $T_1\colon H_1\to H_1$ and $T_2\colon H_2\to H_2$ are called quasisimilar if there exist operators $X\colon H_1\to H_2$ and $Y\colon H_2\to H_1$ such that $T_2X=XT_1$, $YT_2=T_1Y$, and $X$ and $Y$ have zero kernels and dense ranges. It is proved that if two weak contructions $T_1$ and $T_2$ acting on separable spaces $H_1$ and $H_2$ are quasisimilar, then there exists an operator $X\colon H_1\to H_2$ such that $XT_1=T_2$ and the mapping $\mathscr I_X\colon\operatorname{Lat}T_1\to\operatorname{Lat}T_2$, $\mathscr I_XE=\operatorname{clos}XE$, $E\in\operatorname{Lat}T_1$, is a lattice isomorphism. An example is given of two quasisimilar weak contractions such that for any isomorphism $\mathscr I_X$ its inverse is not equal to $\mathscr I_Y$ for an arbitrary (bounded) operator $Y$.