Let $\mathcal H$ be a (separable) Hilbert space on $\mathbb C$.
Suppose $f\in L^2([-\pi,\pi), \mathcal H)$ and $c_n$ is n-th Fourier coefficient of $f$, i.e., $c_n=\frac{1}{2\pi}\int_{-\pi}^\pi e^{-in x}\ f(x)dx.$
Show that if $B:\mathcal H\to\mathcal H$ be a bounded operator, we have $$\sum_{n\in\mathbb Z}\|B (c_n)\|_{\mathcal H}^2=\frac{1}{2\pi}\int_{-\pi}^\pi \|B(f(x))\|_{\mathcal H}^2 dx.$$
I'm having difficulty in showing this.
Somehow I think we can use :
for $g\in \mathcal H$, $(\int f(x)dx, g)_{\mathcal H}=\int(f(x),g)_{\mathcal H}dx$ and $(g, \int f(x)dx)_{\mathcal H}=\int(g,f(x))_{\mathcal H}dx$ hold. (These are already proven.)
e.g., I may have $$\|B(c_n)\|^2 =(B(c_n),B(c_n)) =(B(\int e^{-inx}f(x)dx), B(c_n)) =(\int B( e^{-inx}f(x))dx, B(c_n)) =\int (e^{-inx}B(f(x)), B(c_n)) dx =\int e^{inx}(B(f(x)), B(c_n)) dx $$ (although I don't know whether I can push integral outside of $B$. )
I'm stacked because I don't know how I should do for this identity, how I should handle infinite sum. I also searched solutions but I've not found yet.
Does anyone have idea for proving the identity ? Thanks for the help.