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\}.
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