If two operators commute (eigenvector question)

Suppose [itex]\Omega_1[/itex] and [itex]\Omega_2[/itex] satisfy [itex][\Omega_1,\Omega_2]=0[/itex] and [itex]\Omega = \Omega_1 + \Omega_2[/itex]. If [itex]\Psi_1[/itex] and [itex]\Psi_2[/itex] are eigenvectors of [itex]\Omega_1[/itex] and [itex]\Omega_2[/itex], respectively, don't we know that the (tensor?) product [itex]\Psi = \Psi_1 \Psi_2[/itex] is an eigenvector of [itex]\Omega[/itex]? Also, if the [itex]\Psi_i[/itex] are normalized, isn't [itex]\Psi[/itex] automatically normalized?
Well, I've just worked through this, and I think I've determined the following: If [itex]\Omega_i \Psi_i = a_i \Psi_i[/itex], we have [itex]\Omega \Psi = (a_1 + a_2) \Psi[/itex], regardless of whether [itex][\Omega_1,\Omega_2] = 0[/itex]. Is this true? (I'm assuming the [itex]\Omega_i[/itex] are Hermitean, but even that might not make any difference.)
It seems you are mixing things. Either you are discussing tensor product or not. If you are discussing tensor product and if [itex]
[/itex] and [itex]
[/itex] refer to two different Hilbert spaces, then they automatically commute.

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