martes, 29 de julio de 2014

Café matemático: Extended eigenvalues for Cesàro operators

I just uploaded to the arXiv a preprint of my paper on extended eigenvalues for Cesàro operators. This is joint work with Fernando León-Saavedra (Cádiz), John Petrovic (Michigan) and Omid Zabeti (Iran).
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 $[1,\infty),$ for $C_1$ it is the interval $(0,1],$ and for $C_\infty$ it reduces to the singleton $\{1\}.$