Operators acting in the dual spaces of discrete Cesàro spaces

Abstract

The discrete Cesaro (Banach) sequence spaces $$ {{\text {ces}}}(r), 1< r < \infty ,$$ have been thoroughly investigated for over 45 years. Not so for their dual spaces $$ d (s) \cong ( {{\text {ces}}}(r))', $$ with $$ \frac{1}{s} + \frac{1}{r} = 1 ,$$ which are somewhat unwieldy. Our aim is to undertake a further study of the spaces d(s) and of various operators acting between these spaces. It is shown that $$ d (s) \subseteq d (t)$$ whenever $$ s \le t ,$$ with the inclusion being compact if $$ s< t .$$ The classical Cesaro operator C is continuous from d(s) into d(t) precisely when $$ s \le t $$ and compact precisely when $$ s < t .$$ Moreover, C even maps the larger space $$ {{\text {ces}}}(s)$$ continuously into d(s). This is a consequence of the Hardy–Littlewood maximal theorem and the remarkable property, for each $$ 1< s < \infty ,$$ that $$ x \in \mathbb {C}^{\mathbb {N}} $$ satisfies $$ C (C (| x| )) \in d (s)$$ if and only if $$ C (| x | ) \in d (s).$$ These results are used to analyze the spectrum and to determine the norm and the mean ergodicity of C acting in d(s). Similar properties for multiplier operators are also treated.