Approximation of Entropy Numbers

Abstract

The purpose of this article is to develop a technique to estimate certain bounds for entropy numbers of diagonal operator on \(\ell ^p\) spaces for \(1<p<\infty ,\) which improves the existing bounds. The approximation method we develop in this direction works for a very general class of operators between Banach spaces, in particular, separable Hilbert spaces. As a consequence of this technique we also obtain an alternative proof for the following result for a bounded linear operator T between two separable Hilbert spaces: $$\begin{aligned} \epsilon _k(T)=\epsilon _k(T^*)=\epsilon _k(|T|) \; \text {for each}\; k\in \mathbb N, \end{aligned}$$where \(\epsilon _k(T)\) is the kth entropy number of T.