# Spectral theorem for normal compact operator

Spectral Theorem for Compact Self-Adjoint Operators goes as follows -

Let $V$ be a nonzero Hilbert space, and let $T : V \rightarrow V$ be a compact, self-adjoint operator. Then $V = \overline{\oplus_{\lambda} V_{\lambda} }$. For each $λ \neq 0$, $dim \space V_λ < ∞$, and for each $\epsilon > 0$ , $|\{λ \space | |λ| ≥ \epsilon , dim \space V_λ > 0 \}| < ∞$.

($\lambda$ is the eigenvalue.)

I wanted to ask that is this spectral theorem same as for Compact normal operators ?

• It's the same except that with normal operators, we take $\lambda \in \Bbb C$ as opposed to $\lambda \in \Bbb R$. – Omnomnomnom Nov 22 '16 at 1:16
• The algorithm stays the same, it is the SVD algorithm, and the largest singular vector is guaranteed to be an eigenvector: $v_i = \arg \max_{\|v\|=1} \|T_i v\|$, $\lambda_i = \frac{T_i v_i}{v_i}$ (instead of $\lambda_i = \|T_i v_i\|$) and $T_{i+1} = T_i - \lambda_i v_i v_i^*$ (i.e. ) – reuns Nov 22 '16 at 1:32
• $T$ is compact so it has only a finite number of singular values $> \epsilon$, so that $\lim_{i \to \infty} \|T_i\| = 0$ and the algorithm converges (in norm) to $T = \sum_i \lambda_i v_iv_i^*$ – reuns Nov 22 '16 at 1:34