Positive-definiteness preserved in the quotient space?

Question

Let $H$ be a Hilbert space and $T: H\to H$ be a positive definite, hermitian operator.

Let $N$ be a proper closed subspace of $H$ and $T(N)$ is included in $N$.

Then is the induced operator $T: H/N \to H/N$ also positive definite?

I tried to figure out but things seem trickier than I expected. Could anyone help me?