Extended eigenvalues for Cesàro operators

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 non-zero operator X   such that TX=λXTTX=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 Cesàro operator C0C0, the finite continuous Cesàro operator C1C1 and the infinite continuous Cesàro operator C∞C∞ defined on the complex Banach spaces ℓpℓp, Lp[0,1]Lp[0,1] and Lp[0,∞)Lp[0,∞) for 1