Compatibility Thm HW: Can We Find More Orthon Eigenstates?

[davon806]

Homework Statement

Please see the following,I am confused by the word "only".

Homework Equations

The Attempt at a Solution

I understand that the Compatibility theorem ensures we can find a basis of common eigenfunctions of \hat{A} ,\hat{B}.If each pair of eigenvalues {A_i,B_j} identifies uniquely one vector of the basis,then the set {\hat{A} ,\hat{B}} forms a CSCO.
Here they are asserting \tilde{u_1} , \tilde{u_2} are the only eigenstates of B in the plane.I don't see the reason for that.Could we find more orthonormal eigenstates of B in the plane spanned by the degenerate states?

 

[mfb]

Could we find more orthonormal eigenstates of B in the plane spanned by the degenerate states?

They are not degenerate with respect to B. That's exactly the case (1) discussed there, the two eigenvectors with respect to B are unique. If both eigenvalues of B are the same, see case (2).