Operator Commuting with Diagonalizable Operator

Theorem

Let $H$ be a Hilbert space.

Let $A = \ds \sum_{i \mathop \in I} \alpha_i P_i$ be a diagonalizable operator on $H$.

Let $B \in \map B H$ be a bounded linear operator.

Then the following are equivalent:

$(1): \quad A B = B A$

$(2): \quad$ For all $i \in I$, $\Rng {P_i}$ is a reducing subspace for $B$

where $\Rng {P_i}$ denotes range.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $II.7.5$