Linear Evolution Equations that Involve Products of Commutative Operators

Let $P(\xi _1 , \cdots ,\xi _n )$ be a polynomial of degree $2k$ in $\xi _1 , \cdots ,\xi _n $ with real constant coefficients. Let X be a function space and let $A_1 , \cdots ,A_n $ denote operators in X such that $A_i \cdot A_j = A_j \cdot A_i $; for all $i,j$. This paper treats the problem of representing solutions of the Cauchy problem $u'(t) = P(A_1 , \cdots ,A_n )u(t) $, $t > 0;u(0 + ) = \varphi $ in the following situations: (i) X is a Banach space and the $A_i $ are infinitesimal generators of $C_0 $ groups in X and (ii) X is a space of entire functions and the $A_i $ are derivative operators in X. The results are motivated by elementary operational formulas and applications are given to both well-posed and ill-posed problems.