Well, I'm a little confused. Suppose we have three Hermitian operators
$\widehat A = \widehat A^{\dagger}$
$\widehat B = \widehat B^{\dagger}$
$\widehat C = \widehat C^{\dagger}$.
We know that $[\widehat A, \widehat B] = i \widehat C $ and $[\widehat A,\widehat C] = 0$.
So, we know that $\widehat A f(a) = a f(a) $ because it's its own representation.
But what about $\widehat B f(a) = ?$
I am sure that I can make some statement from above commutation properties. But I can't write smth expect well-known bacics.