Characteristic Equations for the Spectrum of Generators

We propose an abstract framework for the computation of the spectrum (A) of a linear operator A : D(A) X ! X on a Banach space X through a condition in a smaller Banach space X 1. If this space is nite dimensional this yields a characteristic equation for (A). The method is tested for delay, integro-diierential and population equations and is applicable to control and stability questions. The spectrum (A) of the generator A of a strongly continuous semigroup on a Banach space X characterizes various qualitative and, in particular, asymptotic properties of the semigroup. We only mention that the negativity of the spectral bound of A, i.e., s(A) := sup fRe : 2 (A)g < 0 implies uniform exponential stability for eventually norm continuous semigroups on Banach spaces or for positive semigroups on L p-spaces (see Ne], Chapter 3.5). We refer to the monographs Na1] and Ne] where these and many more relations between spectrum and asymptotics are discussed systematically. It thus remains a fundamental task to nd or at least to estimate the spectrum (A) of semigroup generators A. By deenition, this is a problem in the given Banach space. However, in many situations it can be solved by nding the zeros of a certain complex function. More precisely, there exists a function A : C ! C such that the spectral values of A are the zeros of a so-called characteristic equation, i.e., (A) = f 2 C : A () = 0g: Thus the innnite dimensional (linear) problem of computing (A) is reduced to a one-dimensional (non linear) problem. Here are some rst examples for this phenomenon. 1.1 Examples (i) Take the nite dimensional space X := C n and a complex n n-matrix A := (a ij) nn. Then (A) = f 2 C : p A () = 0g for the characteristic polynomial p A () := det(? A). (ii) In the semigroup approach to delay equations one considers the Banach space X := C?1; 0] and the operator Af := f 0 on the domain D(A) := f 2 C 1 ?1; 0] : f 0 (0) = Lf for some continuous linear form L 2 X 0. Then it is well known that (A) = f 2 C : L() ? = 0g; where (s) := e s for s 2 ?1; 0]. These and many more examples lead to the …