Invertible positive maps that are not automorphism

Let $X$ be a real normed vector space with a cone $K\subseteq X$ satisfying either (i) $K$ is closed with non-empty interior or (ii) $K$ has non-zero extremals or (iii) $K$ is closed and $X$ is a Banach space. In this short note, we provide a method to construct an invertible linear map $T\text{\hspace{0.17em}⁣}:X\to X$ such that $T[K]\subseteq K$ but $T^{-1}[K]\subseteq ̸K$. In particular, we show that, for every cone automorphism $S\text{\hspace{0.17em}⁣}:X\to X$, there exists a rank one perturbation of $S$ which is positive and invertible, but does not have a positive inverse. We provide examples from four diverse situations.