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Suppose $Y$ is a closed hyperplane in $X$, so write $X=Y\oplus[x_0]$. Let $y_n$ be a normalized basis of $Y$. Define an operator $S:Y\to Y\oplus[x_0]$ by $Sy_n=\\alpha_ny_n+\beta_nx_0$, for any $n=1,2,\dots$. We can choose $\alpha_n\to 0$ and $\beta_n\to 0$ such that $S$ is compact. Can we find a non-trivial subspace $Z$ of $Y$ such that $S(Z)\subseteq Z$?

Edit: I previously posted a more general question, but I just realized it has a negative answer. This is the concrete example I have.

Suppose $Y$ is a closed hyperplane in $X$, so we can write $X=Y\oplus[x_0]$. Let $y_n$ be a normalized basis of $Y$. Define an operator $S:Y\to Y\oplus[x_0]$ by $Sy_n=\alpha_ny_n+\beta_nx_0$, for any $n=1,2,\dots$. We can choose $\alpha_n\to 0$ and $\beta_n\to 0$ such that $S$ is compact. Can we find a non-trivial subspace $Z$ of $Y$ such that $S(Z)\subseteq Z$?

Edit: I previously posted a more general question, but I just realized it has a negative answer. This is the concrete example I have.

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.

Suppose $Y$ is a closed hyperplane in $X$, so write $X=Y\oplus[x_0]$. Let $y_n$ be a normalized basis of $Y$. Define an operator $S:Y\to Y\oplus[x_0]$ by $Sy_n=\\alpha_ny_n+\beta_nx_0$, for any $n=1,2,\dots$. We can choose $\alpha_n\to 0$ and $\beta_n\to 0$ such that $S$ is compact. Can we find a non-trivial subspace $Z$ of $Y$ such that $S(Z)\subseteq Z$?

Edit: I previously posted a more general question, but I just realized it has a negative answer. This is the concrete example I have.