/sci/ - Science & Math

>>11306002
>>11306026
Better idea:
Set [math]F: \mathbb{C}^4 \rightarrow \mathcal{H}[/math] by [math]F(e_1)=\phi[/math], [math]F(e_2)= A \phi[/math], [math]F(e_3)= B \phi [/math] and [math]F(e_4) = AB \phi [/math] and extend by linearity.
Because [math]A^2=B^2=Id[/math] and [math]AB=-BA[/math], [math]im ~ F[/math] is closed under [math]A[/math] and [math]B[/math].
We can then pull back the operator and the inner product to show that if there is a counterexample, then it exists in [math]\mathbb{C}^4[/math].
Similarly, if it actually works in some case lke "A and B are self adjoint", the trick still works.