Convolution kernel recovered from its L2 operator ($T_h=0\text{ on }{L}^2(\mathbb{R}),h\in L^1⟹h=0$)

An integrable kernel is uniquely determined by its convolution operator on $L^2$. Convolve it with a smooth compactly supported approximate identity ${\eta }_{\epsilon }$, whose members also lie in $L^2$. A zero operator makes $h\ast {\eta }_{\epsilon }=0$, while translation continuity in Lp gives $h\ast {\eta }_{\epsilon }\to h$ in $L^1$, proving the assertion. Since the adjoint operator of convolution by $f$ is convolution by $f^{♯}(s)=\overline{f(-s)}$, this also proves the exact self-adjointness criterion $f=f^{♯}$ almost everywhere, including necessity at the level of kernels.