Carl's inequality for 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 $$\underset{1\le k\le n}{\sup }{k}^{\alpha }{e}_k(T)\le {\gamma }_{\alpha }\underset{1\le k\le n}{\sup }{k}^{\alpha }{c}_k(T)$$ 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.

13 pages, minor changes suggested by referee, accepted for publication in Journal of Functional Analysis