Sub-Markovian semigroups generated by matrix operators (Q1318086)

We characterize generators of sub-Markovian semigroups on \(L^ p(\Omega)\) by a version of Kato's inequality. This will be used to show (under precise assumptions) that the semigroup generated by a matrix operator \({\mathcal A}=(A_{ij})_{1\leq i,j\leq n}\) on \((L^ p(\Omega))^ n\) is sub-Markovian if and only if the semigroup generated by the sum of each row \(A_{i1}+\cdots+ A_{in}\) \((1\leq i\leq n)\) is sub-Markovian. The corresponding result on \((C_ 0(X))^ n\) characterizes dissipative operator matrices.