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Entropy numbers in APTARABOLDITALIC γ ‐Banach spaces
Author(s) -
Kaewtem Thanatkrit
Publication year - 2017
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201600516
Subject(s) - mathematics , banach space , bounded function , finite rank operator , bounded operator , approximation property , c0 semigroup , entropy (arrow of time) , pure mathematics , discrete mathematics , mathematical analysis , physics , quantum mechanics
Let X be a quasi‐Banach space, Y be a γ‐Banach space ( 0 < γ ≤ 1 ) and T be a bounded linear operator from X into Y . In this paper, we prove that the first outer entropy number of T lies between2 1 − 1 / γ∥ T ∥and ∥ T ∥ ; more precisely,2 1 − 1 / γ∥ T ∥ ≤ e 1 ( T ) ≤ ∥ T ∥ , and the constant 2 1 − 1 / γis sharp. Moreover, we show that there exist a Banach space X 0 , a γ‐Banach space Y 0 and a bounded linear operatorT 0 : X 0 → Y 0such that 0 ≠ e k ( T 0 ) = 2 1 − 1 / γ∥ T 0 ∥for all positive integers k . Finally, the paper also provides two‐sided estimates for entropy numbers of embeddings between finite dimensional symmetric γ‐Banach spaces.