# Approximation of Entropy Numbers

## TL;DR

This paper introduces a new technique for estimating entropy numbers of diagonal operators on p-summable sequence spaces, extending to a broad class of operators between Banach spaces and resolving a question about entropy numbers of operators between Hilbert spaces.

## Contribution

Develops a general approximation method for entropy numbers of operators on Banach spaces, including Hilbert spaces, and proves the equality of entropy numbers of an operator and its adjoint in Hilbert spaces.

## Key findings

- Provides bounds for entropy numbers of diagonal operators on p-summable sequences.
- Establishes the equality of entropy numbers of an operator and its adjoint in separable Hilbert spaces.
- Answers a question posed by B. Carl regarding entropy numbers in Hilbert spaces.

## Abstract

The purpose of this article is to develop a technique to estimate certain bounds for entropy numbers of diagonal operator on spaces of p-summable sequences for finite p greater than 1. The approximation method we develop in this direction works for a very general class of operators between Banach spaces, in particular reflexive spaces. As a consequence of this technique we also obtain that the entropy number of a bounded linear operator T between two separable Hilbert spaces is equal to the entropy number of the adjoint of T. This gives a complete answer to the question posed by B. Carl [4] in the setting of separable Hilbert spaces.