Why the Cesàro operator $(Cf)(s) = \frac{1}{s}\int_0^sf(t)dt$ on $L^2(0,\infty)$ is not compact?

Question

By Hardy's inequality, $C$ is bounded. But I find no way to prove it to be noncompact. I was thinking about its adjoint $$ (C^\ast f)(s) = \int_x^\infty\frac{1}{t}f(t)dt $$

But I'm afraid it is equally difficult to show that $C^\ast$ is noncompact. So this approach does not work.

I also attempted to use a classic sequence on the unit ball: $f_n = \chi_{[n,n+1]}$. But it turned out to be a monster. Indeed we have $\lVert f_n \rVert = 1$ and $\lVert f_m-f_n \rVert = \sqrt{2}\delta_{mn}$, which is pretty neat. But computing the norm of $Cf_n$ and $C(f_m-f_n)$ is a disaster: $$ \lVert Cf_n \rVert = \sqrt{2-2n\ln\left( 1 + \frac{1}{n}\right)} \\ \lVert C(f_m-f_n) \rVert =\sqrt{\frac{2n+1}{n+1}-2n\ln\left( 1+\frac{1}{n}\right)+2+\frac{1}{m}-2(m+1)\ln\left( 1+\frac{1}{m}\right)+\ln\frac{m}{n+1}} $$ provided $m>n$ (hopefully my calculation wasn't wrong!). With these being said, is there any accessible way to prove the noncompactness (whether direct or indirect)? I guess I found two inaccessible way (or maybe not as I didn't realise how to use it). Many thanks in advance!

Answer 1

Here is one way to show that the Hardy transform $H:f\mapsto\frac1t\int^t_0f$ as an operator on $L_p(0,\infty)$, where $1<p<\infty$, is not compact. Consider the family of functions $\{\phi_A(x)=A^{1/p}\mathbf{1}_{(0,1/A]}(x): A>0\}$. We have that $$\|\phi_A\|_p=1$$ and $$G_A(x):=(H\phi_A)(x)=\frac1x\int^x_0\phi_A = A^{1/p}\Big(\mathbf{1}_{(0,1/A]}(x) + \frac{1}{Ax}\mathbf{1}_{(1/A,\infty)}(x)\Big)$$ Then, for $A<B$, $$\begin{align} |G_A - G_B|&=\big(B^{1/p}-A^{1/p}\big)\mathbb{1}_{(0,1/B]}(x)+\left|\frac{B^{1-\tfrac1p}}{x}- A^{1/p}\right|\mathbb{1}_{(1/B,1/A]}(x) \\ &\qquad + \big(A^{\tfrac1p -1} -B^{\tfrac1p -1}\big)\frac{1}{x}\mathbb{1}_{(1/A,\infty)}(x)\\ &\geq \big(A^{\tfrac1p -1} -B^{\tfrac1p -1}\big)\frac{1}{x}\mathbf{1}_{(1/A,\infty)}(x) \end{align}$$ whence we conclude that $$\|G_A - G_B\|_p \geq \frac{1}{(p-1)^{1/p}}\left|1-(A/B)^{1-\frac1p}\right|$$ It follows that $\{g_n=\phi_{2^n}:n\in\mathbb{N}\}$ is sequence in unit ball in $L_p(0,\infty)$ for which $\{Hg_n:n\in\mathbb{N}\}$ has no convergent subsequence in $L_p(0,\infty)$.