Understanding the proof of Fuglede's theorem

Question

I am trying to understand this proof of Fuglede's theorem on wikipedia :

Everything was making sense until the last sentence: "Considering the first-order terms in the expansion for small λ, we must have $M^*T = TN^*$." What does that mean?

Answer 1

They have shown that $F(\lambda)\equiv T$. Let’s (carefully!) expand $F$ as a series around $\lambda\approx0$: $$F(\lambda)=T+\lambda M^{\ast}T-\lambda TN^{\ast}+\mathcal{O}(\lambda^2)$$Which implies that: $$\lambda(M^\ast T-TN^\ast)=\mathcal{O}(\lambda^2)$$You can divide away by $\lambda$ and safely take limits as $\lambda\to0$ to conclude $M^\ast T-TN^\ast=0$.

N.B. You can induct to obtain equality of all higher order terms as well, although this gets more complicated and probably less interesting.