Your condition (2) is that $T^*$ is a surjective isomorphism, so $T$ induces a surjective isomorphism on the completion of $X$. For a counterexample, let $T$ be the right shift on $\ell_2(Z)$ restricted to an appropriate dense subspace. What subspace? Well, it must be dense, so throw in the unit vector basis. Throw in some natural vector $y$ which you want to be the limit of $Tx_n$; $y=\sum_{k=1}^\infty 2^{-k!} e_k$ e_k$should be fine. You need for$T$to map the subspace back into itself, so throw in$T^k y$for$k=1,2,...$. Let$X$be the linear span of all the vectors thrown in. Set$x_n= \sum_{k=0}^n 2^{-(k+1)!} e_k$. Then$x_n$converges to a point not in$X$but$Tx_n$converges to$y$. 1 Your condition (2) is that$T^*$is a surjective isomorphism, so$T$induces a surjective isomorphism on the completion of$X$. For a counterexample, let$T$be the right shift on$\ell_2(Z)$restricted to an appropriate dense subspace. What subspace? Well, it must be dense, so throw in the unit vector basis. Throw in some natural vector$y$which you want to be the limit of$Tx_n$;$y=\sum_{k=1}^\infty 2^{-k!} e_k$ should be fine. You need for$T$to map the subspace back into itself, so throw in$T^k y$for$k=1,2,...$. Let$X$be the linear span of all the vectors thrown in. Set$x_n= \sum_{k=0}^n 2^{-(k+1)!} e_k$. Then$x_n$converges to a point not in$X$but$Tx_n$converges to$y\$.