## Extended eigenvalues for Cesàro operators

I just received the news that my paper entitled Extended eigenvalues for Cesàro operators has been accepted for its publication in the Journal of Mathematical Analysis and Applications. This is joint work with Fernando León-Saavedra (Jerez de la Frontera), Srdjan Petrovic (Kalamazoo) and Omid Zabeti (Sistan and Baluchestan). Here is the abstract of the paper.

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 Cesàro operator $C_0,$ the finite continuous Cesàro operator $C_1$ and the infinite continuous Cesàro 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

$\displaystyle{ (C_0f)(n) \colon = \frac{1}{n+1} \sum_{k=0}^n f(k),}$

$\displaystyle{ (C_1f)(x) \colon = \frac{1}{x} \int_0^x f(t)\,dt,}$

$\displaystyle{ (C_\infty f)(x) \colon = \frac{1}{x} \int_0^x f(t)\,dt.}$

It is shown that the set of extended eigenvalues for $C_0$ is the interval $[1,\infty),$ that for $C_1$ it is the interval $(0,1],$ and that for $C_\infty$ it reduces to the singleton $\{1\}.$

## Acerca de Miguel Lacruz

Gijón, Asturias, España, 1963