Is the Stokes operator positive-definite? (I don't understand the argument of the authors in a book that I'm reading)

Question

I'm reading Navier-Stokes Equations and Turbulence. The authors define $$\left\|u\right\|^2:=\left\|\nabla u\right\|_{L^2(\Omega,\:\mathbb R^d)}\;\;\;\text{for }u\in H^1(\Omega,\:\mathbb R^d)$$ and state that the Stokes operator $A$ is positive-define, because $$\langle u,Au\rangle_{L^2(\Omega,\:\mathbb R^d)}=\left\|u\right\|\;\;\;\text{for all }u\in D(A)\color{blue}{\subseteq H^2(\Omega,\:\mathbb R^d)}\;.\tag 1$$ I don't understand this. Shouldn't $(1)$ only imply that $A$ is positive-semidefinite? There are obviously elements $0\ne u\in H^1(\Omega,\:\mathbb R^d)$ with $\nabla u=0$.