On absolutely Cesàro bounded operators

Abstract

We study $p$-absolutely Cesàro bounded operators, with particular emphasis on self-improvement, weighted shifts, and linear dynamics. Our first main result shows that, for $1<p<\infty$, every positive absolutely Cesàro bounded operator on $L^p(Ω)$ is automatically $p$-absolutely Cesàro bounded. We then characterize $q$-absolute Cesàro boundedness of backward shifts on weighted $\ell^p$-spaces for $q\geq p$ and use this characterization to construct examples exhibiting a wide range of possible growth rates of the powers. In the dynamical direction, we obtain new obstructions to absolute and strong Cesàro boundedness; in particular, no strongly Cesàro bounded operator on a nonzero Banach space is chaotic or upper frequently hypercyclic. We also introduce Cesàro ratio-boundedness and compare it with absolute Cesàro boundedness. Finally, a general Baire-category principle yields a genericity theorem for the failure of $p$-absolute Cesàro boundedness.