Extended eigenvalues for Cesàro operators

Abstract

A complex scalar λ 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 T X = λXT. Such an operator X is called an extended eigenoperator of T corresponding to the extended eigenvalue λ. The purpose of this paper is to give a description of the extended eigenvalues for the discrete Ces`aro operator C0, the finite continuous Ces`aro operator C1 and the infinite continuous Ces`aro operator C∞ defined on the complex Banach spaces ℓ p , Lp [0, 1] and L p [0, ∞) for 1 < p < ∞ by the expressions (C0f)(n): = 1 n + 1 Xn k=0 f(k), (C1f)(x): = 1 x Z x 0 f(t) dt, (C∞f)(x): = 1 x Z x 0 f(t) dt. It is shown that the set of extended eigenvalues for C0 is the interval [1, ∞), for C1 it is the interval (0, 1], and for C∞ it reduces to the singleton {1}.