Let A = B(l²), x be the unilateral shift operator on l², defined by T(a₁,a₂,...) = (0,a₁,a₂,...), and y = T^*. Then TT^*(a₁,a₂,¼) = T(a₂,a₃,¼) = (0,a₂,a₃,¼) and T^*T(a₁,a₂,¼) = T^*(0,a₁,a₂,¼) = (a₁,a₂,¼). Hence sp(T^*T) = {1} but 0 Î sp(TT^*) (since (TT^*)(1,0,0,¼) = (0,0,¼)).