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.