A criterion for property \(C\) (Q1310384)

Let \({\mathcal L}_ j\) be formal partial differential operators in \(\mathbb{R}^ n\), \(n\geq 2\), with constant coefficients, \(N({\mathcal L}_ j)= \{u: {\mathcal L}_ j u= 0\}\). The authors say that the pair \(\{{\mathcal L}_ 1,{\mathcal L}_ 2\}\) has property \(C\) if and only if for any compact domain \(D\subset \mathbb{R}^ n\), the set \(\{u_ 1u_ 2\} \forall u_ j\in N({\mathcal L}_ j)\) is complete in \(L^ 2(D)\). They give a necessary and sufficient condition for the pair \(\{{\mathcal L}_ 1,{\mathcal L}_ 2\}\) to have property \(C\).