%0 Journal Article
%T Carl's inequality for quasi-Banach spaces
%A Aicke Hinrichs
%A Anton Kolleck
%A Jan Vybiral
%J Mathematics
%D 2015
%I arXiv
%X We prove that for any two quasi-Banach spaces $X$ and $Y$ and any $\alpha>0$ there exists a constant $c_\alpha>0$ such that $$ \sup_{1\le k\le n}k^{\alpha}e_k(T)\le c_\alpha \sup_{1\le k\le n} k^\alpha c_k(T) $$ 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, which closes a gap in the argument of Donoho in his seminal paper on compressed sensing.
%U http://arxiv.org/abs/1512.04421v1