Another proof of the iniectivity of a linear operator

Question

Let $g(x)= \chi_{[-\frac{1}{2}, \frac{1}{2}]}(x) $, and $ T \colon L^2(\mathbb{R}) \longrightarrow L^2 (\mathbb{R})$ , $Tf= g \star f$. I was asked to prove that $T$ is injective, and I succedeed using the Fourier transform. My question is: is there another proof?

Answer 1

You can probably start with the smooth case $f \in C_0(\mathbb{R})$, and write down explicitly $$\int_{-1/2}^{1/2} f(x-y)\, dy =0 \quad\hbox{for (almost) every $x \in \mathbb{R}$}.$$ This should imply rather easily that $f$ vanishes identically. Then remember that a generic $f \in L^2(\mathbb{R})$ can be approximated by continuous functions with compact support.