On certain Opial-type results in Ces\`aro spaces of vector-valued functions

TL;DR

This paper investigates how Opial-type properties in Banach spaces extend to Cesàro spaces of vector-valued functions and sequences, establishing conditions under which these properties are preserved or transferred.

Contribution

It demonstrates that Opial and uniform Opial properties in Banach spaces imply similar properties in associated Cesàro spaces of functions and sequences, extending known geometric properties.

Findings

Opial properties are preserved in Cesàro spaces under certain conditions.

Results apply to both function spaces over [0,1] and sequence spaces.

Provides new insights into the geometric structure of Cesàro spaces.

Abstract

Given a Banach space $X$, we consider Ces\`aro spaces $\text{Ces}_p(X)$ of $X$-valued functions over the interval $[0,1]$, where $1\leq p<\infty$. We prove that if $X$ has the Opial/uniform Opial property, then certain analogous properties also hold for $\text{Ces}_p(X)$. We also prove a result on the Opial/uniform Opial property of Ces\`aro spaces of vector-valued sequences.