On the Fredholm property of linear elliptic operators in \(\mathbb R^n\) (Q1778295)

The author considers a linear partial operator-differential expression \(P\) acting on vector-functions on \(\mathbb R^n\) with values from a Hilbert space \(X\). Suppose first that \(\dim X<\infty\). As an operator from the Sobolev space \(H^m(\mathbb R^n,X)\) to \(L_2(\mathbb R^n,X)\), \(P\) is shown to be Fredholm if and only if \(P\) and its formal adjoint satisfy coercivity estimates. For the case where \(\dim X=\infty\), this property is proved for \(\Phi_e\)-operators, that is if, for any bounded sequence \(u_j\in H^m\) such that \(Pu_j\) is locally convergent in \(L_2\), the sequence \(Pu_j\) contains a subsequence locally convergent in \(H^m\). On the other hand, if \(\dim X<\infty\) and the leading coefficients are constant, then the \(\Phi_e\)-property of \(P\) is equivalent to its ellipticity.