I am currently trying to prove that every bounded $T:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$ which commutes with translations must be a Fourier multiplier. I've seen a few posts that deal with this question and there are some good answers which are convincing to me.
That being said, the exercise I was given, was to prove as a first step that such an operator must commute with any convolution operator with a function $g\in L^1(\mathbb{R}^d)$.
The weird thing is - I am unsure as to what does this commutation mean. The notation $g*(Tf)(x)$ is clear but what does $T(g*f)(x)$ even say?
I though about "skipping" this and trying to work with the Fourier transform of $T$, but got nothing usefull so far (I understand that there is a better way to prove the theorem, but I'm really curious to know what was the intention of said exercise).