Spectrum of Hardy operators on $L^p[0,1]$

Question

Let $Hf(x)=\frac{1}{x}\int_0^x f(t)dt$ denote the well-known (continuous) Hardy (someone called Cesáro) operators. For $1<p<+\infty$, it can shown that $H$ is bounded on $L^p[0,+\infty)$ with operator norm $\frac{p}{p-1}$. It is also true that $H$ is bounded on $L^p[0,1]$ with the same norm. For the spectrum, when $p=2$, Brown, Halmos and Shields (1965) proved that $H$ are both respectively unitary equivalent to certain shift operators and hence deduce that $$\sigma(H,L^2[0,\infty))=\{\lambda: |\lambda-1|=1\}$$ and $$\sigma(H,L^2[0,1])=\{\lambda:|\lambda-1|\leq 1\}.$$ Tow years later, Boyd proved that $\sigma(H,L^p[0,+\infty)=\{\lambda: |\lambda-\frac{p}{2(p-1)}|=\frac{p}{2(p-1)}\}$. It is reasonable to guess $$\sigma(H,L^p[0,1])=\{\lambda: |\lambda-\frac{p}{2(p-1)}|\leq\frac{p}{2(p-1)}\}.$$

But I have not find references up to now. Is anyone know some references about this? Of course one can show the spectrum of $H$ on $L^p[0,1]$ directly.

Answer 1

Above question is positively answered by this paper.