Compact inverse and adjoint of a coercive elliptic operator ($(I-\lambda K)u=Kf,K=L^{-1}$)

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