# Significance and application of Riesz Decomposition Theorem

The Riesz Decomposition Theorem in Operator Theory is given as: Let $a \in \mathcal{A}$ (for unital Banach algebra $\mathcal{A}$) Suppose $\sigma(a) = \sigma_1 \cup \sigma_2$ where $\sigma_1 \cap \sigma_2 = \emptyset$. Then:

1. $\exists$ non-trivial idempotents $E_1~,E_2 \in \mathcal{A}$ such that $E_1 + E_2 = 1$.

2. If $\mathcal{A} \subset \mathcal{L(X)}$ ($X$ Banach space) then $E_1X, ~E_2X$ are closed subspaces invariant for $a$ and $E_1X \oplus E_2X = X$.

3. If $a_k = aE_k$ then $\sigma(a_k) = \sigma_k$ (for $k=1,2$) and for any $f \in Hol(a)$ we have $$f(a_k) = f(a)E_k.$$

Question: What is the significance of these results, how is it most commonly used? At the moment it seems like an arbitrary collection of results, but as I understand it is an important result in operator and spectral theory. Also, what are the applications in quantum mechanics?

Thanks.

• Did you mean $\sigma_1 \cap \sigma_2 = \emptyset$? – DisintegratingByParts Nov 8 '16 at 17:06
• @TrialAndError Yeah it was a mistake, I have corrected it now. – Alex Nov 8 '16 at 17:14
• The subsets $\sigma_1$ and $\sigma_2$ should be closed I guess? – Mike F Nov 8 '16 at 17:19
• @MikeF Yeah. In the notes I was working with, the sets are not given as closed, but I think that was omitted by mistake. – Alex Nov 8 '16 at 17:26
• Yes, the disconnected components should be closed. – DisintegratingByParts Nov 8 '16 at 18:37