Fractional Powers of Operators Corresponding to Coercive Problems in Lipschitz Domains

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$, $n\ge2$, and let $L$ be a second-order matrix strongly elliptic operator in $\Omega$ written in divergence form. There is a vast literature dealing with the study of domains of fractional powers of operators corresponding to various problems (beginning with the Dirichlet and Neumann problems) with homogeneous boundary conditions for the equation $Lu=f$, including the solution of the Kato square root problem, which arose in 1961. Mixed problems and a class of problems for higher-order systems have been covered as well.
We suggest a new abstract approach to the topic, which permits one to obtain the results that we deem to be most important in a much simpler and unified way and cover new operators, namely, classical boundary operators on the Lipschitz boundary $\Gamma=\partial\Omega$ or part of it. To this end, we simultaneously consider two well-known operators associated with the boundary value problem.