So here is my problem,
Let $\rho \in C^\infty (\mathbb{R}^n,R)$ with $\rho\geq 0$, $\rho(x)= 0 \; \forall \|x\|\geq 1$ and $\int_{\mathbb{R}^n}\rho(x)dx=1$. Further, consider the linear map $K_f:L^p\rightarrow L^p$ given by $K_f(g)=f*g$ the convolution of f and g.
Now consider $\rho_m(x):=m^N\rho (mx).$
I have to decide whether $1\leq p < \infty$: $\|K_{\rho_m}-Id\|_{L^p\rightarrow L^p}\rightarrow 0$? is true?
I have already shown that $K_{\rho_m}(g) \rightarrow g$, as $m \rightarrow \infty$ in $L^p$ for every $g \in L^p$.
Can somebody give me a hint?
Thanks
No, $K_{\rho_m}$ does not converge to the identity in the operator norm. No matter how large $m$ is, you can find a function $g$ such that $\rho_m*g$ is substantially different from $g$. Hint: consider scaling $g$ in the same way: $g_m(x) = m^N g(mx)$. Details after the break
Fix some $g$ such that $\|\rho*g-g\|_p\ne 0$; e.g., characteristic function of a ball. Use the change of variables to show that $\rho_m*g_m = (\rho*g)_m$, where all subscripts refer to the scaling parameter. Conclude that $$\|\rho_m*g_m-g_m\|_p = m^{N(p-1)}\|\rho*g-g\|_p$$ and therefore $$\frac{\|\rho_m*g_m-g_m\|_p}{\|g_m\|_p} = \frac{ \|\rho*g-g\|_p}{\|g\|_p}$$ which is nonzero and independent of $m$.