Consider the Sobolev space on the torus $\mathbb{T}=\mathbb{R}/2\pi \mathbb{Z}$ given by $H^m(\mathbb{T})$ is the $L^2$-functions such that the Fourier coefficients satisfy the decay rate $\sum_{n\in\mathbb{Z}} |\hat{f}(n)|^2(1+n^2)^m<\infty$ (this is the square of the norm). Suppose $A: H^m(\mathbb{T})\rightarrow L^2(\mathbb{T})$ is bounded and linear, and there exists $B:L^2(\mathbb{T})\rightarrow H^m(\mathbb{T})$ such that $AB-I=E$ and $BA-I=F$ where $I$ is the identity and $E,F$ are compact. Suppose also that $(Af,g) = (f,Ag)$ whenever $f,g\in H^m(\mathbb{T})$, where $(,)$ is the $L^2$ inner product. The goal is to show that $A-\lambda I: H^m(\mathbb{T})\rightarrow L^2(\mathbb{T})$ is invertible when $\lambda\notin \mathbb{R}$.
As a hint, it is given that $E^*$ maps $L^2$ into $H^m$ and $(Bf,g)=(f,Bg)$ for all $f,g\in L^2(\mathbb{T})$. I was able to show that $(Ef,g)=(f,Fg)$ whenever $g\in H^m$ and $f\in L^2$ but I wasn't sure how to make use of this. I'm not sure how to mimic the proof of the spectral theorem for compact operators into this case.