Let $\mathcal X$ be the Banach space of $L^1$ functions on some probability space, $\mathcal Y$ be some other Banach space, $T:\mathcal X\to \mathcal Y$ be some surjective continuous linear map, $\mathcal X_+$ be the set of all elements of $\mathcal X$ with a nonnegative version (a closed convex cone), and $\mathcal Y_+:=T(\mathcal X_+)$ (a convex cone).

Question:

- Is $\mathcal Y_+$ necessarily closed in $\mathcal Y$?
- If not, are there nice, easily verifiable conditions on $\mathcal Y_+$ that are sufficient for it to be closed?
- Do either of these answers change if I'm willing to assume that $\mathcal Y$ is itself a closed subspace of some $L^1$ space, and $T$ is positive?