Abstract
The main purpose of the paper is to investigate the spectra of the q-Cesaro operator C-q, where 0<q<infinity, on Banach spaces. For 0<q<1, the method of the proof is to exhibit a relationship between the operator C-q and a generalized difference operator Delta(ab). Furthermore, for 1<q<infinity, it is shown that the operator C-q is compact on certain Banach spaces. In this case, the spectra of C-q are completely determined.