Let $\mathbb{E}$ and $\mathbb{Y}$ are Euclidean spaces, $K\subseteq\mathbb{E}$ is a proper cone and $A\colon\mathbb{E}\to\mathbb{Y}$ is a linear transformation, what is the relation between $A(K)^\ast$ and $A(K^\ast)$ ?
If there is no relation, is there any extra assumption on $A$ that would give me some result of this kind?
There is a clear relation if $A$ is invertible.
Let $\mathbb{E}$ and $\mathbb{Y}$ be Euclidean spaces, let $K\subseteq\mathbb{E}$ and $L\subseteq\mathbb{Y}$ both be proper cones, and let $A\colon\mathbb{E}\to\mathbb{Y}$ be an invertible linear transformation. Then: $$\cdot (A^\ast(L))^\ast=A^{-1}(L^\ast)$$ $$\cdot A(K)^\ast=(A^\ast)^{-1}(K^\ast) $$
PROOF: To show the first:
$x\in (A^\ast(L))^\ast \iff \langle A^\ast(x),y\rangle\geq 0 \text{ for each } y\in L\iff \langle x,A(y)\rangle\geq 0\text{ for each } y\in L\iff A(x)\in L^\ast\iff x\in A^{-1}(L^\ast).$
To show the second :
$x\in A(K)^\ast\iff \langle x,A(y)\rangle\geq0\text{ for each }y\in K \iff \langle A^\ast(x),y\rangle\geq 0\text{ for each } y\in K \iff A(x)\in K^\ast\iff x\in (A^\ast)^{-1}(K^\ast).$