Suppose $Y\subseteq X$ is a proper, infinite dimensional, closed subspace of a Banach space $X$ and $S:Y\to X$ is a compact operator. Does there exists $Z\subseteq Y$ such that $S(Z)\subseteq Z$?

In general the answer is no, as Read has constructed a strictly singular operator without invariant subspaces, and any strictly singular has a compact restriction to some infinite dimensional subspace. Read's construction is on $l_2$-sum of James $p$-spaces, so the space $X$ is non-reflexive in his case.

My question is, are there any known conditions on $X$ (i.e. reflexive, Hilbert), on $Y$ (i.e. $Y$ complemented, $Y$ finite co-dimensional), or on $T$ that ensures the above problem has a positive answer.