Evolution semigroups and sums of commuting operators: A new approach to the admissibility theory of function spaces

This payer is concerned with conditions for the admissibility of a translat ion invariant function space M with respect to a well posed linear evolutio n equation du/dt = Au + f(t), t is an element of R (*). We propose a new ap proach to this problem by considering the sum of two commuting operators - d/dt : = - D-M and the operator of multiplication by A on M. On the one han d, the closure of this operator is the infinitesimal generator of the so-ca lled evolution semigroup associated with (*). On the other hand, the genera tor G of this semigroup relates a mild solution u of (*) to the forcing ter m f by the rule Gu = -f. Consequently, various spectral criteria of the typ e sigma(D-M) boolean AND sigma(A) = circle divide for the admissibility of the function space M with respect to (*) can be proved in an elegant manner . Moreover, they can be naturally extended to general classes of differenti al equations, including higher order and abstract functional differential e quations. Applications and examples are provided to illustrate the obtained results. (C) 2000 Academic Press.