# Some New Bounds on the Entropy Numbers of Diagonal Operators

## TL;DR

This paper establishes new bounds on the entropy numbers of diagonal operators between different p spaces, analyzes their optimality, and provides examples demonstrating the improvements over existing results.

## Contribution

It introduces new upper bounds for entropy numbers of diagonal operators between p spaces and investigates their optimality under various decay conditions.

## Key findings

- New upper bounds for entropy numbers of diagonal operators.
- Optimality proven for fast decaying sequences when p<q.
- Examples illustrating the bounds' applicability.

## Abstract

Entropy numbers are an important tool for quantifying the compactness of operators. Besides establishing new upper bounds on the entropy numbers of diagonal operators $D_\sigma$ from $\ell_p$ to $\ell_q$, where $p\not=q$, we investigate the optimality of these bounds. In case of $p<q$ optimality is proven for fast decaying diagonal sequences, which include exponentially decreasing sequences. In case of $p>q$ we show optimality under weaker assumption than previously used in the literature. In addition, we illustrate the benefit of our results with examples not covered in the literature so far.