I am stuck with Step 2 of the Strichartz's estimate. My question is actually a continuation of a topic which has been raised some time ago and it could be seen here Technical question about Strichartz estimate's proof..
I would really appreciate if someone could help me. I actually struggle with the third equality, namely how the identity below is determined:
$\int_{0}^{t}\int_{0}^{t} (\mathcal{T}(t-s)f(s),\mathcal{T}(t-\sigma)f(\sigma))_{L^2} d\sigma ds=\int_{0}^{t}\int_{0}^{t} (f(s),\mathcal{T}(s-\sigma)f(\sigma))_{L^2} d\sigma ds$.
If I am not wrong, I need to prove that
$\int\mathcal{T}(t-s)f(s)\overline{\mathcal{T}(t-\sigma)f(\sigma)}=\int f(s)\overline{\mathcal{T}(s-\sigma)f(\sigma)}$,
where the $L^2$ inner product is defined by $(u,v)_{L^2}$=Re $\int_{\Omega}u(x)\overline{v(x)}dx$. And it is pretty much where I am stuck as to me this relation is either incorrect or I am too stupid to see it :(
Thanks in advance!