Let $X$ be a Hilbert space with a positive self dual cone $K$. Then a bounded operator $T$ is called
- positive if $T(K) \subseteq K$
- Strongly positive if $T\left(\mathrm{int}\left(K\right)\right) \subseteq K \setminus \{0\} $.
I verified that if $K$ is self-dual, i.e. $K^{*}=K$, then $T$ is positive implies $T^{*}$ is positive. I don't know whether such implication is true for strong positivity. Please help me tho check whether the following holds or not:
"If $K$ is a self dual cone, then $T$ is strongly positive implies $T^{*}$ is strongly positive."
I was just able to prove that $T^{*}x\neq 0$ for $x\in \mathrm{int}(K)$, but not able to proceed further.