Self-improvement for absolutely Cesáro bounded operators

Abstract

We prove that every $p$-absolutely Cesàro bounded operator with constant $K>1$ is $q$-absolutely Cesàro bounded for $1\le q<pK/(K-1)$. This range is optimal, even for positive weighted shifts. On Banach spaces of type $s\in (1,2]$, we prove that every absolutely Cesàro bounded operator is $q$-absolutely Cesàro bounded for every $1\le q<s$, with a logarithmic estimate at $q=s$. We also prove the polynomial growth bound $\Vert T^n\Vert =O(n^{1/s-\epsilon })$ for some $\epsilon >0$. On $L^p$-spaces, $1<p<\infty$, we extend the implication from absolute Cesàro boundedness to $p$-absolute Cesàro boundedness from positive to individually eventually positive operators. Finally, for every $1\le p<\infty$, we construct a $p$-absolutely Cesàro bounded operator $T$ on ${\ell }^1$ such that $\sigma (T)\subseteq \mathbb{D}\cup \{1\}$ but ${\mathrm{limsup}}_{n\to \infty }\Vert T^n(I-T)\Vert =\infty$. We also observe that the construction yields a single operator which is $p$-absolutely Cesàro bounded for every $1\le p<\infty$.

arXiv comment21 pages