Let's call a map closed if it takes any closed subset to a closed subset.
I'm wondering if there are some "standard" examples of closed (in this sense) linear maps on a Banach space, other than the zero map or a linear homeomorphism. More precisely:
Question: What is an example of a bounded linear map $T\colon X\to X$, where $X$ is a Banach space, that takes closed sets to closed sets, other than $T=0$ or a homeomorphism?
As it was noted in the comments (and easy to see), a nonzero closed linear map of Banach spaces is necessarily injective. Hence, if $T: X\to X$ is a closed bounded nonzero linear operator, then the image of $T$ is a closed subspace $Y$ of $X$ and $T: X\to Y$ is a bounded invertible map. Conversely, if $T: X\to X$ is a bounded injective linear map with closed image, then $T$ is a closed map. There are easy examples of such maps. For instance, take $X=\ell_p, 1\le p\le\infty$, and let $$ T: (x_1, x_2,...)\mapsto (0, x_1, x_2, ...) $$ be the shift map. Then $T$ is a closed bounded injective (but not surjective) map.