Carl's inequality for quasi-Banach spaces

TL;DR

This paper extends Carl's inequality, relating entropy and Gelfand numbers, to the setting of quasi-Banach spaces, providing a new fundamental inequality in this broader context.

Contribution

It proves a new version of Carl's inequality for quasi-Banach spaces, which was previously known only for Banach spaces.

Findings

Established the inequality for all linear bounded operators between quasi-Banach spaces.

Demonstrated the inequality's validity for the first time in the quasi-Banach setting.

Provides a key tool for analyzing operator approximation in quasi-Banach spaces.

Abstract

We prove that for any two quasi-Banach spaces $X$ and $Y$ and any $\alpha>0$ there exists a constant $\gamma_\alpha>0$ such that $$ \sup_{1\le k\le n}k^{\alpha}e_k(T)\le \gamma_\alpha \sup_{1\le k\le n} k^\alpha c_k(T) $$ holds for all linear and bounded operators $T:X\to Y$. Here $e_k(T)$ is the $k$-th entropy number of $T$ and $c_k(T)$ is the $k$-th Gelfand number of $T$. For Banach spaces $X$ and $Y$ this inequality is widely used and well-known as Carl's inequality. For general quasi-Banach spaces it is a new result.