Interpolation of Ces{\`a}ro sequence and function spaces

TL;DR

This paper investigates the interpolation properties of Cesàro sequence and function spaces, establishing conditions under which these spaces serve as interpolation spaces and highlighting cases where they do not.

Contribution

It provides new results on when Cesàro spaces are interpolation spaces, extending understanding of their structure in functional analysis.

Findings

Cesàro sequence spaces interpolate between different p-values for 1 < p0 < p1 ≤ ∞.

Cesàro function spaces interpolate similarly under certain conditions.

Cesàro space on [0, 1] is not an interpolation space between Cesàro spaces at p=1 and p=∞.

Abstract

The interpolation property of Ces{\`a}ro sequence and function spaces is investigated. It is shown that $Ces_p(I)$ is an interpolation space between $Ces_{p_0}(I)$ and $Ces_{p_1}(I)$ for $1 < p_0 < p_1 \leq \infty$ and $1/p = (1 - \theta)/p_0 + \theta /p_1$ with $0 < \theta < 1$, where $I = [0, \infty)$ or $[0, 1]$. The same result is true for Ces{\`a}ro sequence spaces. On the other hand, $Ces_p[0, 1]$ is not an interpolation space between $Ces_1[0, 1]$ and $Ces_{\infty}[0, 1]$.