Adjoint of an integration operator anchored at zero ($Tf(t)={\int }_0^{t}f(s)ds$)

On $L^2(-a,a)$, the adjoint operator is $T^{\ast }g(s)={\int }_s^{a}g(t)dt$ for $s>0$, and $T^{\ast }g(s)=-{\int }_{-a}^{s}g(t)dt$ for $s<0$. These formulas follow by reversing the triangular integration order separately on the two half-intervals. Consequently $(T+T^{\ast })g$ is constant on each half, with values ${\int }_0^{a}g$ and $-{\int }_{-a}^{0}g$. Applying this adjoint to a real weight transfers a weighted integral of an anchored function to an integral against its derivative.