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 ). 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 ).
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  and ).