Interior of set of positive operators

Question

In interior of the set of positive operators the question is about the topology of $\mathcal{P}(\mathcal{H}):=\{A\in \mathcal{L}(\mathcal{H})\mid \langle Ax,x\rangle \geq 0\}$, namely if the interior is empty, if the ambient space is $\mathcal{L}(\mathcal{H})$ for a Hilbert space $\mathcal{H}$. I was wondering, how a change of ambient space might change the interior. For this, I want to compute $\partial \mathcal{P}(\mathcal{H})$ in $\mathcal{L}_s(\mathcal{H})$, or even better, I want to show, that $\mathcal{F}_s^+(\mathcal{H}):=\{T\in\mathcal{F}(\mathcal{H}):\sigma(T)>0\}$ is in fact the interior of $\mathcal{P}(\mathcal{H})$. One can actually show, that if $0\in T$ for a selfadjoint Fredholm operator $T$, then $T\in \partial \mathcal{P}(\mathcal{H})$, but I fail to show the converse inclusion. Or is there even a counterexample?