Polynomial operator matrices as semigroup generators: The \(2\times 2\) case (Q1106993)

Many systems of linear evolution equations can be written as a single equation (*) \(\dot u(t)={\mathcal A}_ U(t)\), where u is a function with values in a product space \(E^ n\) and \({\mathcal A}=(A_{ij})_{n\times n}\) is a matrix whose entries \(A_{ij}\) are linear operators on E. In order to prove the well-posedness of (*) one shows that \({\mathcal A}\) generates a strongly continuous semigroup on \(E^ n\). We consider the case where the \(A_{ij}\) are polynomials \(p_{ij}(A)\) with respect to a single (unbounded) operator A on E and restrict our attention to the case of \(2\times 2\) matrices. In the main result we characterize those matrices generating a strongly continuous semigroup on \(E^ n\).