# Unitarily-equivalent representations

Representations $\pi_1$, $\pi_2$ of a group (algebra, ring, semi-group, cf. Representation of a group) $X$ in Hilbert spaces $H_1$, $H_2$, satisfying the condition
$$U\pi_1(x)=\pi_2(x)U$$
for a certain unitary operator $U\colon H_1\to H_2$ and all $x\in X$. Cf. Intertwining operator.