I should prove that $$\exp \left ( \frac{\pi}{2\hbar^2}(L_x^2 + L_y^2) \right )$$ is not a unitary operator. Where $L$ is the total angular momentum of a 2-particle system ($L = L_A + L_B$ for the particles $A$ and $B$).
My (undergraduate) definition of unitary operator is:
$U$ is a unitary operator if $UU^+ = U^+U=I$, where $I$ is the identity operator and $U^+$ is the adjoint of $U$
I have tried using that $$e^U = \sum_k^{\infty}\frac{U^k}{k!}$$ but without any success. I also wrote down some properties of exponential matrices but I really don't know how to proceed here.
EDIT:
I have noticed there is a typo in a subsequent part of the exercise ($L_z^2$ swapped with $L_z$). Could it be possible that I should consider $$\exp \left ( \frac{\pi}{2\hbar}(L_x + L_y) \right )?$$
EDIT2: It was $\hbar^2$ and not $\hbar$. And I posted an answer.