POLYNOMIAL OPERATOR MATRICES AS SEMIGROUP GENERATORS - THE 2X2 CASE

Many systems of linear evolution equations can be written as a single equation $$\dot u(t)=\s A u(t),\eqno(*)$$ where $u$ is a function with values in a product space $E^n$ and $\s A =(A_{ij})_{n\times n}$ is a matrix whose entries $A_{ij}$ are linear operators on $E$. In order to prove the well-posedness of $(*)$ one shows that $\s A$ generates a strongly continuous semigroup on $E^n$. In this paper 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 $2\times2$ matrices.