Proof explanation: tensor product of operators

Question

Let $E$ be an infinite-dimensional complex Hilbert space, $E\otimes E$ be the Hilbert space tensor product and

We recall the following theorem:

Stochel's Theorem: Let $A_1, A_2,B_1, B_2\in \mathcal{L}(E)$ be non-zero operators. The following conditions are equivalent:

• $A_1\otimes B_1=A_2\otimes B_2$.

• There exists $z\in \mathbb{C}^*$ such that $A_1 =zA_2$ and $B_1= z^{-1}B_2$.

This is the proof from the paper

I don't understand why the operators $A_1, A_2,B_1, B_2$ must be non-zero? I think it is sufficient to assume that $A_1$ and $B_1$ are non-zero.

Answer 1

If $A_1$ and $B_1$ are nonzero, then necessarily $A_2$ and $B_2$ are nonzero, so it is the same. Many authors are not looking to optimize the hypotheses of theorems to the bare logical minimum that makes them work. Here, it is not possible to have $A_1,B_1$ nonzero and $A_2=0$ or $B_2=0$.