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 $$Q_A(u)={\int }_{{\mathbb{R}}^N}A{\mathrm{\nabla }}^{s}u(x)\cdot {\mathrm{\nabla }}^{s}u(x)dx,$$ where ${\mathrm{\nabla }}^s$ denotes the Riesz fractional gradient, we establish necessary and sufficient conditions such that $Q_A$ defines a Dirichlet form. In strike contrast with the local case $s=1$, positive definiteness of a constant matrix is not sufficient for the Markov property when $0<s<1$. We also describe the associated symmetric Markov semigroup and its $L^p$ realizations, prove the existence of a smooth nonnegative heat kernel and establish a bounded $H^{\mathrm{\infty }}$ functional calculus.