Find the adjoint operator of $T:L^2((0,1))\rightarrow \mathbb{R}^2$ $x\mapsto (\int^1_0sx(s)ds, \int^1_0s^2x(s)ds)$
My attempt was to look at $\langle Tf,g\rangle = \langle f, T^{\ast}g\rangle$
\begin{align*}
\langle Tf,g\rangle = \int_0^1 Tf(t)g(t)dt = \int^1_0 (\int_0^1 sf(s)ds,\int_0^1 s^2f(s)ds) g(t) dt
\end{align*}
I tried to proceed with integrating by parts but that just leads me to…
\begin{align*}
\int^1_0 (\int_0^1 sf(s)ds,\int_0^1 s^2f(s)ds) g(t) dt =\int^1_0 (F(1) -\int_0^1 F(s)ds, F(1)-\int^1_0 sF(s)ds) g(t) dt
\end{align*}
I don't think that's useful. I want to get to the form $\int_0^1 f(t) … dt$ so that I can rewrite it to $\langle f, T^{\ast}g\rangle$.
Any suggestions?
Best Answer
You got the defintion of the adjoint wrong.
$T^{*}$ is an operator from the dual of $\mathbb R^{2}$ which can be identified with $\mathbb R^{2}$ itself into $L^{2}(0,1)$. $T^{*}(a,b) x=(a,b)Tx=a\int_0^{1} sx(s)ds+b \int_0^{1} s^{2}x(s)ds$. Hence $T^{*}(a,b)=as+bs^{2}$.