CAUCHY-PROBLEMS FOR POLYNOMIAL OPERATOR MATRICES ON ABSTRACT ENERGY SPACES

For a $n\times n$ polynomial matrix $M$ and a linear operator $A$ on some Banach space $E$ we previously defined the polynomial operator matrix $\sA :=M(A)$ on the product space $\sE :=E^n$, but needed quite strong conditions on $M$ and $A$ to ensure that $\sA$ is the generator of a strongly continuous semigroup on $\sE$. In this paper we will show that it is possible to find a restriction $\sAG$ of $\sA$ to an abstract ``energy space" $\sEG$ such that $\sAG$ is a generator on $\sEG$ under much less restrictive assumptions on $M$ and $A$.