Order spectrum of the Cesàro operator in Banach lattice sequence spaces

Abstract

The discrete Cesaro operator C acts continuously in various classical Banach sequence spaces within $$ {\mathbb {C}}^{{\mathbb {N}}}.$$ For the coordinatewise order, many such sequence spaces X are also complex Banach lattices [eg. $$c_0, \ell ^p $$ for $$ 1 < p \le \infty , $$ and $$ {{\text {ces}}}(p)$$ for $$ p \in \{ 0 \} \cup ( 1, \infty )$$]. In such Banach lattice sequence spaces, C is always a positive operator. Hence, its order spectrum is well defined within the Banach algebra of all regular operators on X. The purpose of this note is to show, for every X belonging to the above list of Banach lattice sequence spaces, that the order spectrum $$ \sigma _\mathrm{o} (C)$$ of Ccoincides with its usual spectrum $$ \sigma ( C)$$ when C is considered as a continuous linear operator on the Banach space X.