Compute the adjoint of an operator in $L^2[-1,1]$

Question

Consider a bounded linear operator $A:L^2[-1,1]\to L^2[-1,1]$ given by $Af(t)=\int_{-1}^0f(s)ds+(\int_{-1}^1f(s)ds)t^2$.

Question: Compute the adjoint of the operator $A$.

I know we need to calculate $\langle Af,g\rangle=\langle f, A^*g\rangle$, but I have troubles of calculating $\langle f, A^*g\rangle$ through the calculation of integral. Could anyone help me? Thanks in advance!

Answer 1

Hint: Verify that $\int Af(t)g(t)dt=\int_{-1}^{1} f(t) [ I_{(-1,0)}(t)\int_{-1}^{1}g(s)+\int_{-1}^{1}g(s)s^{2}]ds$. Hence $A^{*}g(t)=I_{(-1,0)}(t)\int_{-1}^{1}g(s)ds+\int_{-1}^{1}g(s)s^{2}ds$.

Here $I_A$ is defined by $I_A(x)=1$ if $x \in A$ and $0$ if $x \notin A$.