The solution of the Kato problem in two dimensions

We solve, in two dimensions, the "square root problem of Kato". That is, for $L\equiv -\operatorname{div} (A(x)\nabla)$, where $A(x)$ is a $2\times 2 $ accretive matrix of bounded measurable complex coefficients, we prove that $L^{1/2}\colon L^2_1(\mathbb{R}^2) \rightarrow L^2(\mathbb{R}^2)$.