Completeness of Taylor sequence spaces

Talebi, Gholamreza

doi:10.22103/jmmr.2026.26959.1949

Completeness of Taylor sequence spaces

Articles in Press

Document Type : Research Paper

Author

Gholamreza Talebi

Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

10.22103/jmmr.2026.26959.1949

Abstract

We prove that the Taylor sequence space $t_p^\theta$ is complete for every $1\le p\le\infty$ and $0\le\theta<\frac12$. Although $t_p^\theta$ is naturally isometrically embedded into $\ell_p$, the surjectivity of the Taylor operator is not known \emph{a priori}; therefore, completeness does not follow immediately. The main step is to show that the inverse Taylor matrix defines an operator on $\ell_p$, which holds precisely for $0\le\theta<\frac12$. It follows that the Taylor operator is an isometric isomorphism from $t_p^\theta$ onto $\ell_p$, yielding the desired completeness. As consequences, we deduce several geometric properties of $t_p^\theta$, including reflexivity, the Banach--Saks property, and the fixed point property for nonexpansive mappings.

Keywords

Taylor sequence spaces‎

‎completeness‎

‎uniform rotundity‎

Subjects

Analysis

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