Entropy numbers in $\gamma$-Banach spaces

TL;DR

This paper establishes sharp bounds for the first outer entropy number of operators between quasi-Banach and b3-Banach spaces, revealing precise constants and constructing examples that attain these bounds.

Contribution

It provides the exact value range for the first outer entropy number in b3-Banach spaces and constructs operators that achieve these bounds, advancing understanding of entropy numbers in quasi-Banach settings.

Findings

The first outer entropy number e_1(T) is bounded between 2^{1-1/b3} b2;Tb2 and b2;Tb2.

The constant 2^{1-1/b3} is proven to be sharp.

Examples of operators with all entropy numbers equal to 2^{1-1/b3} b2;Tb2 are constructed.

Abstract

Let $X$ be a quasi-Banach space, $Y$ a $\gamma$-Banach space $(0<\gamma \leq 1)$ and $T$ a bounded linear operator from $X$ into $Y$. In this paper, we prove that the first outer entropy number of $T$ lies between $2^{1-1/\gamma}\|T\|$ and $\|T\|$; more precisely, $2^{1-1/\gamma}\|T\| \leq e_1(T) \leq \|T\|,$ and the constant $2^{1-1/\gamma}$ is sharp. Moreover, we show that there exist a Banach space $X_0$, a $\gamma$-Banach space $Y_0$ and a bounded linear operator $T_0:X_0 \rightarrow Y_0$ such that $0 \neq e_k(T_0) = 2^{1-1/\gamma}\|T_0\| $ for all positive integers $k.$ Finally, the paper also provides two-sided estimates for entropy numbers of embeddings between finite dimensional symmetric $\gamma$-Banach spaces.