= Adjoint of an integration operator anchored at zero
{title2=$Tf(t)=\int_0^t f(s)ds$}
On $L^2(-a,a)$, the <adjoint operator> is $T^*g(s)=\int_s^ag(t)dt$ for $s>0$, and $T^*g(s)=-\int_{-a}^sg(t)dt$ for $s<0$. These formulas follow by reversing the triangular integration order separately on the two half-intervals. Consequently $(T+T^*)g$ is constant on each half, with values $\int_0^ag$ and $-\int_{-a}^0g$. Applying this adjoint to a real weight transfers a weighted integral of an anchored function to an integral against its derivative.
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