= Compact inverse and adjoint of a coercive elliptic operator
{title2=$(I-\lambda K)u=Kf,\quad K=L^{-1}$}
A coercive <Dirichlet boundary condition>[zero-Dirichlet] weak form gives a <bounded> solution map from $L^2$ into $H_0^1$. Composing with the compact <compact embedding>[Sobolev embedding] into $L^2$ gives the compact inverse $K$. Transposing the <bilinear form> produces its Hilbert-space adjoint inverse $K^*$. The Fredholm compatibility condition for a shifted equation is <orthogonality> of $f$ to the homogeneous adjoint solutions. Indeed $(Kf,w)=(f,w)/\lambda$ when $K^*w=w/\lambda$, and the zero spectral parameter is excluded at resonance by coercivity.
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