Is $A$ an positive operator if $JA$ is a positive operator?

Question

Let us assume that we have a complex, separable and $\mathbb{Z}/2\mathbb{Z}-$ graded space Hilbert space $\mathcal{H}.$ So $\mathcal{H} = \mathcal{H}_0 \oplus \mathcal{H}_1$. Let us denote $P_0$ and $P_1$ ortoghonal projections onto $\mathcal{H}_0$ and $\mathcal{H}_1$ respectively. Let $J = P_1 - P_0$. Of course $J$ is bounded linear operator. Moreover $J$ is self-adjoint and $J^2 = I$. From last two we have that $J$ is an unitary operator. Let us assume that we have bounded, linear opeator $A:\mathcal{H} \to \mathcal{H},$ such that $\langle JAh,h\rangle_\mathcal{H} \ge 0$ and $A \ne J$. So $JA$ is an positive operator and, since $\mathcal{H}$ is complex Hilbert space also a self-ajoint operator. Operator $J$ is clearly not positive because $\langle Jh,h\rangle_\mathcal{H} = \langle P_1h,h\rangle_\mathcal{H} - \langle P_0h,h\rangle_\mathcal{H} = \langle P_1h,P_1h\rangle_\mathcal{H} - \langle P_0h,P_0h\rangle_\mathcal{H} = \Vert P_1h \Vert_\mathcal{H}^2 - \Vert P_0h \Vert_\mathcal{H}^2$ so by taking $h \in \mathcal{H}_0$ we obtain some negative value. Since $(JA)^{\dagger} = JA = A^{\dagger}J^{\dagger} = A^{\dagger}J.$ So $JA = A^{\dagger}J$. My question is if, with some additional assumptions, we can obtain that $A$ is a positive operator.