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Section 8.6 Summary of Spectral Theory

\begin{align*} A\amp:\amp X\to X, \qquad \mbox{linear}\\ \sigma(A) \amp=\amp \sigma_p(A)\cup \sigma_c(A) \cup \sigma_r(A)\\ \sigma_p(A) \amp=\amp \{\mbox{eigenvalues}\}\\ \sigma_c(A) \amp=\amp \{R_\lambda(A) \mbox{ exists with dense domain, but is unbounded}\}\\ \sigma_r(A) \amp=\amp \{R_\lambda(A) \mbox{ exists, but the domain is not dense}\} \end{align*}
  • Neumann series \(\displaystyle (I-A)^{-1} = \sum_{k=0}^\infty A^k \)
  • spectral radius \(\displaystyle r_\sigma(A) = \sup_{\lambda\in \sigma(A)} |\lambda| \leq \|A\|\)
  • resolvent set \(\rho(A) = \CC\backslash \sigma(A)\text{.}\) \(\rho(A)\) is open.

{\bf Spectral Theorems:}
  1. \(A\) symmetric on \(H\)-space: \(\sigma_p(A)\subset \RR, \sigma_c(A) \subset\RR\text{;}\) eigenvectors are orthogonal.
  2. \(A\) compact on \(H\)-space. \(\sigma(A)=\sigma_p(A)\text{;}\) \(\lambda\in \sigma_p(A)\) has finite multiplicity. \(\sigma_p(A)\) is a discrete set with possible limit point at \(0\text{.}\)
  3. \(A\) compact and self-adjoint: In this case we have the properties of Item 1. and Item 2. plus the spectral representation
    \begin{equation*} Au = \sum_{i=1}^\infty \lambda_i(\psi_i, u) \psi_i, \qquad \{\psi_i\}_i \mbox{ orthonormal basis of eigenvectors}. \end{equation*}