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]).