Extended eigenvalues for Ces\`aro operators

A complex scalar $\lambda$ is said to be an extended eigenvalue of a bounded linear operator $T$ on a complex Banach space if there is a nonzero operator $X$ such that $TX= \lambda XT.$ Such an operator $X$ is called an extended eigenoperator of $T$ corresponding to the extended eigenvalue $\lambda.$ The purpose of this paper is to give a description of the extended eigenvalues for the discrete Ces\`aro operator $C_0,$ the finite continuous Ces\`aro operator $C_1$ and the infinite continuous Ces\`aro operator $C_\infty$ defined on the complex Banach spaces $\ell^p,$ $L^p[0,1]$ and $L^p[0,\infty)$ for $1 < p <\infty$ by the expressions \begin{align*} (C_0f)(n) \colon & = \frac{1}{n+1} \sum_{k=0}^n f(k),\\ (C_1f)(x) \colon & = \frac{1}{x} \int_0^x f(t)\,dt,\\ (C_\infty f)(x) \colon & = \frac{1}{x} \int_0^x f(t)\,dt. \end{align*} It is shown that the set of extended eigenvalues for $C_0$ is the interval $[1,\infty),$ for $C_1$ it is the interval $(0,1],$ and for $C_\infty$ it reduces to the singleton $\{1\}.$