Let $N: \mathcal{A} \rightarrow \mathcal{B}$ be a positive linear map between two Hilbert spaces, $\mathcal{A}$ and $\mathcal{B}$. Define the adjoint map $N^\dagger$ as the linear map which satisfies for any $x\in \mathcal{A}$ and $y\in\mathcal{B}$
$$\langle y, N(x)\rangle = \langle N^\dagger(y), x\rangle$$
Under what conditions is the map $N^\dagger N$ a strictly positive map? That is, when do we have the condition below for any $x$?
$$\langle x, N^\dagger N x\rangle > 0$$
As Theo Bendit suggests, we have $N^\dagger N > 0$ if and only if $N$ is injective.
If $N$ is injective then for $x \ne 0$ we have $Nx \ne 0$ so $$\langle x, N^\dagger Nx\rangle = \langle Nx,Nx\rangle = \|Nx\|^2 > 0$$
Conversely, if $N^\dagger N > 0$, then for $x \ne 0$ we have $$\|Nx\|^2 = \langle Nx,Nx\rangle = \langle x, N^\dagger Nx\rangle > 0$$ so $Nx \ne 0$. It follows that $N$ is injective.