Operator Commuting with Diagonalizable Operator

Theorem
Let $H$ be a Hilbert space.

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

Let $B \in B \left({H}\right)$ be a bounded linear operator.

Then the following are equivalent:


 * $(1): \qquad AB = BA$
 * $(2): \qquad$ For all $i \in I$, $\operatorname{ran} P_i$ is a reducing subspace for $B$

where $\operatorname{ran}$ denotes range.