The problem of completeness of the linear span of eigenvectors and generalized eigenvectors is an important topic in the theory of nonselfadjoint operators with a long and interesting history (see [3] and [4]). In the Hilbert space setting the results mainly concern operators that are close to selfadjoint operators. From the results in the previous section it follows that we are not only interested in completeness of the linear span of eigenvectors and generalized eigenvectors, but also in noncompleteness. The reason is that noncompleteness of the linear span of eigenvectors and generalized eigenvectors of the operator G given by (4) implies that equation (2) and (3) have nontrivial small solutions. Most of the known results discuss sufficient conditions for completeness and the question of noncompleteness is, in general, more difficult. In this way the theory of functional differential equations yields new completeness problems for nonselfadjoint operators and some `old' results turn out to admit unexpected new generalizations. Not only for autonomous equations as described above, but also for periodic equations.
To further illustrate this, we consider the following simple model problem
where b is a continuous periodic function with period 1 and
x is prescribed by a continuous function on the interval [s-1,s].
A function 
is called an elementary solution
if 
, p(t) = p(t+1) and x(t) satisfies equation (7)
for 
.
In that case 
is an eigenvalue of the monodromy operator
(or time-one map)
which acts as a bounded operator on C[-1,0]. Furthermore, any solution
of (7) can be approximated by a linear combination of elementary solutions
if and only if the space spanned
by the system of eigenvectors and generalized eigenvectors of T is dense
in C[-1,0]. Or equivalently, if and only if the operator 
on the Hilbert space 
has a complete set of eigenvectors and generalized eigenvectors, i.e., the
span of the space spanned by the system of eigenvectors and generalized
eigenvectors of is dense in . It is
known that completeness occurs for T or if and only if
b does not change sign (see [7]).
If the period q of the periodic function b is different from the delay, the representation of the monodromy operator (or the time-q map) is more difficult. The analysis of such more complicated operators is subject of current investigations.