The Stinespring-Wittstock theory for variable exponential completely bounded maps

Mykola Ivanovich Yaremenko

Full PDF

abstract_carpathian_2026_42_1_115-122

https://doi.org/10.37193/CJM.2026.01.08

Published on 30 September 2025

Issue no: Vol 42/2026 no. 1 Tags: Radon measure, \textbf{\textit{$p\left(\cdot \right)$-}}completely bounded map, Bochner integral, Bochner theory, Stinespring-Wittstock theorem, factorization, relative compact

Description

Abstract.

We establish the fundamental factorization theorem for operators between Banach spaces under the condition of the variable exponent $p\left(\cdot \right)$ -completely boundedness. Assuming $X,\quad Y$ and $\tilde{X},\quad \tilde{Y}$ be separable Banach spaces, we show that the variable exponent $p\left(\cdot \right)$ -completely bounded mapping \textbf{\textit{ $\Lambda \; :\quad S\to LB\left(\tilde{X},\; \tilde{Y}\right),$ $S\subset LB\left(X,\; Y\right)$ }} can be presented in the form $\Lambda \left(a\right)=\theta _{2} \hat{\pi }\left(a\right)\theta _{1}$ , where $\theta _{1} \; :\quad \tilde{X}\to V/N$ , $\theta _{2} \; :\quad \breve{V}/\breve{N}\to \tilde{Y}$ , and $\hat{\pi }\left(a\right)\; :\quad V/N\to \breve{V}/\breve{N}$ such that $\left\| \theta _{1} \right\| \left\| \theta _{2} \right\| \le \left\| \Lambda \right\| _{p\left(\cdot \right)-cb}$ , the reverse is also true.