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e.lipnowski
e.lipnowski
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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?

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 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?

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?
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e.lipnowski
e.lipnowski
  • 537
  • 2
  • 13

Closedness of linear image of positive L1 functions

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 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?