  The linearized Korteweg-De Vries equation can be written as a Hamilton-like
system. However, the Hamilton energy depends on the time, and is a nonsymmetric
operator on $L^2({\bf R})$. By performing suitable unitary transforms on the
Hamilton energy, we can reduce this operator into one that is not independent
on the time but nonsymmetric. In this study, we consider the $L^2$-stability
issues and smoothing estimates for this operator, and prove that it has no
eigenvalues.
