Restrictions of bounded linear operators: Closed range

Let $T$ be a bounded linear operator on a Banach space $X,$ and let $W$ be a subspace of $X$ which is a Banach space and $T-$ invariant. Denote by $T_{W}$ the restriction of $T$ to $W.$ This paper explores the questions:

$\circ$ If the range of $T$ is closed, under what conditions is the range of $T_{W}$ closed?

$\circ$ If the range of $T_{W}$ is closed, under what conditions is the range of $T$ closed?