Entropy numbers in $\gamma$-Banach spaces
Thanatkrit Kaewtem

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 be a quasi-Banach space, a -Banach space and a bounded linear operator from into . In this paper, we prove that the first outer entropy number of lies between and ; more precisely, and the constant is sharp. Moreover, we show that there exist a Banach space , a -Banach space and a bounded linear operator such that for all positive integers Finally, the paper also provides two-sided estimates for entropy numbers of embeddings between finite dimensional symmetric -Banach spaces.
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Taxonomy
TopicsAdvanced Banach Space Theory
