Discrete spectrum of a symmetric elliptic Dirichlet operator

Spectral data of $K_{\mu }$. By [F1] and [F2] the nonzero eigenvalues $\nu$ of $K_{\mu }$ are real and of finite multiplicity; positivity gives $\nu >0$ for each of them, and they accumulate only at $0$. The eigenspaces $E_{\nu }$ are finite dimensional and pairwise orthogonal, and their union spans a dense subspace: its closed span is $(\ker K_{\mu }{)}^{\perp }=L^2(\mathrm{\Omega })$.

Translation. Let $\nu >0$ be an eigenvalue of $K_{\mu }$ with eigenvector $e$. Since $K_{\mu }e\in H_0^1(\mathrm{\Omega })$ and $K_{\mu }e=\nu e$, we have $e\in H_0^1(\mathrm{\Omega })$; then [F3] shows $a(e,v)=({\nu }^{-1}-\mu )(e,v{)}_{L^2}$ for every $v\in H_0^1(\mathrm{\Omega })$, so $({\nu }^{-1}-\mu ,e)$ is a weak eigenpair with finite-dimensional eigenspace equal to the $\nu$-eigenspace of $K_{\mu }$; conversely every weak eigenpair $(\lambda ,u)$ arises this way from $\nu =(\lambda +\mu {)}^{-1}$, and $\lambda ={\nu }^{-1}-\mu$. In particular the two eigenvalue lists correspond bijectively, and each weak eigenvalue is real and of finite multiplicity.

Enumeration. By [F2] the closed span of the nonzero eigenspaces is all of $L^2(\mathrm{\Omega })$, which is infinite dimensional by [F1]. Since each eigenspace is finite dimensional, there must be infinitely many nonzero eigenvalues; compactness gives at most countably many. By [F4], together with Countable Choice, choose an orthonormal basis of each eigenspace $E_{\nu }$; their union $\{e_j{\}}_{j\ge 1}$ is an orthonormal family whose closed span is $L^2(\mathrm{\Omega })$ by step 1.1, hence a Hilbert basis of $L^2(\mathrm{\Omega })$ with $e_j\in H_0^1(\mathrm{\Omega })$ and $K_{\mu }{e}_j={\nu }_j{e}_j$, where the eigenvalues ${\nu }_j>0$ are listed in decreasing order with multiplicity, so that ${\nu }_j\downarrow 0$. Set ${\lambda }_j:={\nu }_j^{-1}-\mu$; then ${\lambda }_j$ is nondecreasing and tends to $+\mathrm{\infty }$, and step 2.1 gives $a(e_j,v)={\lambda }_j(e_j,v{)}_{L^2}$ for every $v\in H_0^1(\mathrm{\Omega })$, equivalently $L{e}_j={\lambda }_j{e}_j$ by Symmetric elliptic weak eigenpairs.

Claim 2 and the lower bounds. Every weak eigenvalue occurs in the list $\{{\lambda }_j\}$ by step 2.1, and each listed ${\lambda }_j$ is a weak eigenvalue; eigenspaces for distinct eigenvalues are $L^2$-orthogonal by [F2], since they are eigenspaces of $K_{\mu }$ for distinct $\nu$. Finally ${\lambda }_j={\nu }_j^{-1}-\mu >-\mu$ because ${\nu }_j>0$, and Garding's inequality gives $a(u,u)\ge -\beta \mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2$ for every $u\in H_0^1(\mathrm{\Omega })$.

Conclusion. Steps 3.1 and 4.1 establish all three assertions: the list $\{{\lambda }_j\}$, the orthonormal basis $\{e_j\}$ with the weak eigenrelations and the operator form $L{e}_j={\lambda }_j{e}_j$, the completeness of the eigenvalue list with finite multiplicities and orthogonality of distinct eigenspaces, and the lower bounds ${\lambda }_j>-\mu$ and $a(u,u)\ge -\beta \mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2$. No Hilbert basis of $\ker K_{\mu }$ was selected, because $K_{\mu }$ is injective by [F1]; only orthonormal bases of the finite-dimensional nonzero eigenspaces were chosen.