In case *G* is a nonpositive selfadjoint operator on a Hilbert space *H*, the solution of

can be written down in terms of the spectral resolution of *G* as the integral

The asymptotic behaviour of this integral is like , where denotes the infimum of those values of for which . It follows from this asymptotic description that no solution of (5) different from the zero solution, tends to zero in norm faster than some exponential function . A similar result holds for an operator *G* that is close to a selfadjoint operator (see [5]). However, for arbitrary nonselfadjoint operators *G* equation (5) can have solutions that tend to zero faster than any exponential function . In fact such small solutions can exist for functional differential equations. For example, the system

has nontrivial solutions which tend to zero faster than any exponential (see [1]).

More generally, necessary and sufficient conditions for the existence of small solutions for equation (2) can be deduced from the spectral data of the corresponding operator *G*. In order to illustrate this we give some more definitions.

A sequence of vectors in *C* with is called a *Jordan chain* of *G* at if

Note that is an eigenvector of *G*, the vectors are called *generalized eigenvectors* at . A Jordan chain of *G* gives rise to a special solution of (3). If is a Jordan chain of *G* at , then

is a solution of (3) for all *t*. So if the linear space spanned by all eigenvectors and generalized eigenvectors is dense in *C* then each solution can be approximated by a linear combination of solutions of the form (6) (in a sense that can be made precise). In this case we say that the operator *G* has a complete span of eigenvectors and generalized eigenvectors. It is known that the operator *G* given by (4) has a complete span of eigenvectors and generalized eigenvectors if and only if the asymptotic behaviour of

where *n* denotes the dimension of the system (the size of the matrix function ). Furthermore equation (3) has nontrivial small solutions if and only if the corresponding operator *G* does not have a complete span of eigenvectors (see [2] and [6]).

Fri Mar 20 16:01:06 MET 1998