An statement concerning finite rank projections

Question

Let $H$ be a Hilbert space. Let $T$ be an isometric operator on $H$. Suppose that $P$ is a finite rank projection with $PT^nP=PT^n$ for every $n\geq1$.

Q. Can we conclude that $P(H)\subseteq \bigcap_{n\geq1} T^n(H)$

Answer 1

This is not true. Take $H=l^2(\mathbb N)$, $T$ right-shift, $P$ projection onto $span(e_1)$, i.e. $$ Px=(x_1,0,0,\dots). $$ Then $PT=0$, $e_1\in P(H)$, but $e_1\not\in T^n(H)$ for all $n$. Hence, the inclusion is not even true if intersection is replaced by union.