$X$ be Banach space , $T \in \mathcal B(X)$ be an open map , $Y$ be a closed linear subspace of $X$ ; is the restriction of $T$ on $Y$ an open map?

Question

Let $X$ be a Banach space , let $T$ be a continuous open linear map from $X$ to $X$ , let $Y$ be a closed linear subspace of $X$ , then is $T_o$ , the restriction of $T$ on $Y$ , is an open map ? Please help. Thanks in advance

Answer 1

No it is not true. Consider $Vect(e_n,n\in N)$ endowed with $\|\|_{\infty}$. Let $T$ defined by $T(e_n)=e_{n-1},n>0, T(e_0)=0$. $T$ is surjective, thus the open mapping theorem implies that $T$ is open. Let $Y=Vect(e_0)$. The restriction of $T$ to $Y$ is not open.