I have this example on a textbook but it does not seem to be right.
Example: If A,B are unitary operators and AB=BA,prove that A and B have same eigenvalues.
2026-04-01 22:14:08.1775081648
Do two unitary operators that commute have same eigenvalues?
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They do not necessarily have the same eigenvalues. Consider the operators $I$ and $-I$, which definitely commute, and are both self-adjoint and square to $I$. However, the eigenvalues of $I$ are just $1$, whereas the eigenvalues of $-I$ are just $-1$. You might be looking for the statement that commuting diagonalizable operators are simultaneously diagonalizable; that is, that they have the same eigenvectors.