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 $TX=\lambda XT\mathrm{.}$ Such an operator $X$ is called an extended eigenoperator of $T$ corresponding to the extended eigenvalue $\lambda \mathrm{.}$ 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 $C_{\mathrm{\infty }}$ defined on the complex Banach spaces ${\mathrm{\ell }}^p,$ $L^p[0,1]$ and $L^p[0,\mathrm{\infty })$ 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 $[1,\mathrm{\infty }),$ for $C_1$ it is the interval $(0,1],$ and for $C_{\mathrm{\infty }}$ it reduces to the singleton $\{1\}\mathrm{.}$