Uniqueness implies existence for the elliptic Dirichlet problem

Existence and uniqueness. If the homogeneous problem is trivial then $N=\{0\}=\ker (I-\mu K_{\mu })$ by [F2], so alternative (2) of [F1] is excluded; by the dichotomy of [F1] alternative (1) holds. Hence for every $f\in L^2(\mathrm{\Omega })$ the weak problem has exactly one solution $u\in H_0^1(\mathrm{\Omega })$. The triviality of the adjoint homogeneous problem is not needed for this conclusion, and the two triviality hypotheses are equivalent by The elliptic kernel and cokernel are finite dimensional.

Bounded solution map. Under the hypothesis $N=\{0\}$ the operator $A:=I-\mu K_{\mu }$ is injective on $L^2(\mathrm{\Omega })$, so [F3] makes it boundedly invertible there; since $u$ is a weak solution with datum $f$ exactly when $Au=K_{\mu }f$ by [F2], the unique solution is $u=A^{-1}{K}_{\mu }f$. The operator $K_{\mu }$ commutes with $A=I-\mu K_{\mu }$, hence also with $A^{-1}$, so $u=K_{\mu }{A}^{-1}f$. Both factors in this expression are bounded linear operators, with $K_{\mu }$ mapping into $H_0^1(\mathrm{\Omega })$ by [F4], so $f\mapsto u$ is a bounded linear operator from $L^2(\mathrm{\Omega })$ to $H_0^1(\mathrm{\Omega })$; the expression is independent of the admissible shift because $u$ is.

Conclusion. Steps 1.1 and 2.1 give existence, uniqueness and the bounded solution map for every $f\in L^2(\mathrm{\Omega })$, with the explicit representation $u=K_{\mu }(I-\mu K_{\mu }{)}^{-1}f$; no compactness is used beyond the Fredholm alternative inherited from $\mu K_{\mu }$, and the Axiom of Choice supplies the hypotheses of the Rellich compactness and abstract Fredholm suppliers.