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Compact and limited operators
Author(s) -
Bachir Mohammed,
Flores Gonzalo,
TapiaGarcía Sebastián
Publication year - 2021
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.201900329
Subject(s) - mathematics , lipschitz continuity , differentiable function , compact space , banach space , bounded function , hilbert space , pure mathematics , bounded operator , operator (biology) , rank (graph theory) , normed vector space , extension (predicate logic) , discrete mathematics , class (philosophy) , mathematical analysis , combinatorics , biochemistry , chemistry , repressor , artificial intelligence , gene , programming language , computer science , transcription factor
Let T : Y → X be a bounded linear operator between two real normed spaces. We characterize compactness of T in terms of differentiability of the Lipschitz functions defined on X with values in another normed space Z . Furthermore, using a similar technique we can also characterize finite rank operators in terms of differentiability of a wider class of functions but still with Lipschitz flavour. As an application we obtain a Banach–Stone‐like theorem. On the other hand, we give an extension of a result of Bourgain and Diestel related to limited operators and cosingularity.