The semigroup generated by linear fractional divergence form operators in $L^p(\mathbb{R}^N)$: the constant coefficient case

Let $0<s<1$ and $A$ be a constant symmetric positive definite matrix. Given the quadratic form $$ Q_A(u)=\int_{\R^N} A\nabla^s u(x)\cdot \nabla^s u(x)\,dx, $$ where $\nabla^s$ denotes the Riesz fractional gradient, we establish necessary and sufficient conditions such that $Q_A$ defines a Dirichlet