Let $\rho_\epsilon$ be a mollifier that has support in $B(0,\epsilon)$, define the operator$$T_\epsilon (f)=\rho_\epsilon*f-f,$$for every $f\in L^2(\mathbb R^d)$, can we prove or disprove that the operator norm$$||T_\epsilon||=\sup_{||f||_{L^2}=1} ||T_\epsilon f||_{L^2}\rightarrow 0,$$as $\epsilon \rightarrow 0$? Is there a way to find the function that maximizes this operator norm?
We know that these operators are uniformly bounded (independent of $\epsilon$), by using Fourier transform:$$||T_\epsilon||\leq ||\hat\rho-1||_{L^\infty}^{1/2}<\infty,$$and we also know for each fixed $f\in L^2$, $||T_\epsilon f||_{L^2}\rightarrow 0$. However, I could not find any reference on the convergence speed in terms of $||f||_{L^2}$ and $\epsilon$.
