If we look at the hermitian adjoint of the operator \( T^{\lambda}_{\mu} \), we see via the commutation relations that \( (T^{\lambda}_{\mu})^{\dagger} \) is not an irreducible tensor, that is $$ [J_{\pm}, (T^{\lambda}_{\mu})^{\dagger}]= -\sqrt{(\lambda\pm \mu)(\lambda\mp \mu+1)}(T^{\lambda}_{\mu\mp 1})^{\dagger}, $$ and $$ [J_{z}, (T^{\lambda}_{\mu})^{\dagger}]=-\mu (T^{\lambda}_{\mu})^{\dagger}. $$ The hermitian adjoint \( (T^{\lambda}_{\mu})^{\dagger} \) is not an irreducible tensor. As an example, consider the spherical harmonics for \( l=1 \) and \( m_l=\pm 1 \). These functions are $$ Y^{l=1}_{m_l=1}(\theta,\phi)=-\sqrt{\frac{3}{8\pi}}\sin{(\theta)}\exp{\imath\phi}, $$ and $$ Y^{l=1}_{m_l=-1}(\theta,\phi)=\sqrt{\frac{3}{8\pi}}\sin{(\theta)}\exp{-\imath\phi}, $$