I have a question concerning the commutator of translation invariant operators on $L^2(\mathbb{R})$. Recall that $S:L^2(\mathbb{R})\to L^2(\mathbb{R})$ is translation invariant if $Su_t=u_tS$ for all $t\in\mathbb{R}$ where $(u_tf)(x)=f(x+t)$. The space of such operators is well-studied due for example to the work of Hörmander. The questions is not the following: Let $$ \mathcal{A}:=\{S:L^2(\mathbb{R})\to L^2(\mathbb{R}):\ S\ \text{is translation-invariant}\}, $$ what is $\mathcal{A}'$ (the commutant of $\mathcal{A}$ inside $L^2(\mathbb{R})$)? Is there any explicit description of this space and if so do you have some literature sources for such results? Thank you very much in advance for your help.

I have a question concerning the commutator of translation invariant operators on $L^2(\mathbb{R})$. Recall that $S:L^2(\mathbb{R})\to L^2(\mathbb{R})$ is translation invariant if $Su_t=u_tS$ for all $t\in\mathbb{R}$ where $(u_tf)(x)=f(x+t)$. The space of such operators is well-studied due for example to the work of Hörmander. The question is the following: Let $$ \mathcal{A}:=\{S:L^2(\mathbb{R})\to L^2(\mathbb{R}):\ S\ \text{is translation-invariant}\}, $$ what is $\mathcal{A}'$ (the commutant of $\mathcal{A}$ inside $L^2(\mathbb{R})$)? Is there any explicit description of this space, and if so, do you have some literature sources for such results? Thank you very much in advance for your help.