Crescents and the real variable Cesaro operator

Abstract

This paper explores a version of the classical Cesàro integral operator for the Lebesgue space $L^p(0, 1)$ where we discuss its norm, spectral properties, cyclicity, and invariant subspaces. The spectrum of the Cesàro operator will be a crescent domain whose geometry depends on $p$. An important tool will be semigroups of weighted composition operators on $L^p(0, 1)$.