These equations are out of Sakurai and Napolitano Modern Quantum Mechanics.
I'm trying to show that $T_q^{(2)}$--which is defined as the bilinear of two vector operators $V_i$ and $W_j$--transforms as a second-rank spherical tensor.
$$T_{\pm2}^{(2)}=U_{\pm1}V_{\pm1}$$
$$T_{\pm1}^{(2)}=\frac{U_{\pm1}V_0+U_0V_{\pm1}}{\sqrt2}$$
$$T_{0}^{(2)}=\frac{U_{+1}V_{-1}+2 \, U_{0}V_{0}+U_{-1}V_{+1}}{\sqrt6}$$
I think I need to show that these vectors transform properly under rotations, but I'm not exactly sure. Any help would be appreciated.
