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## Completeness problems

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.

Next: References Up: Nonselfadjoint Problems Previous: Small solutions

J.H.M.Dassen
Fri Mar 20 16:01:06 MET 1998