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\}.

Acerca de Miguel Lacruz

Gijón, Asturias, España, 1963
Esta entrada fue publicada en Research. Guarda el enlace permanente.

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