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Lifting some approximation properties from a dual space $X’$ to the Banach space $X$

Volume 257 / 2021

Ju Myung Kim, Silvia Lassalle, Pablo Turco Studia Mathematica 257 (2021), 287-294 MSC: Primary 46B28, 47L20; Secondary 46B45. DOI: 10.4064/sm190826-20-2 Published online: 5 October 2020

Abstract

For a fixed Banach operator ideal $\mathcal A $, we characterize $\mathcal A $-compact sets (in the sense of Carl and Stephani) that are determined by $c_0$ via the Banach composition ideal $\mathcal A \circ \mathfrak K_{\infty }$, with $\mathfrak K_{\infty }$ the Banach ideal of Fourie and Swart. This characterization allows us to relate $\mathcal K _{\mathcal A} $-approximation properties on a Banach space and $\mathcal K _{\mathcal B} $-approximation properties on its dual space, where $\mathcal A $ and $\mathcal B $ are ideals linked by some classical procedures. These approximation properties have been widely studied in several papers in the last years.

Authors

  • Ju Myung KimDepartment of Mathematics
    Sejong University
    Seoul 05006, Korea
    e-mail
  • Silvia LassalleDepartamento de Matemática
    Universidad de San Andrés Vito Dumas 284
    (B1644BID) Victoria, Buenos Aires, Argentina
    and
    IMAS – CONICET
    e-mail
  • Pablo TurcoIMAS – UBA – CONICET Pabellón I
    Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires (1428) Buenos Aires, Argentina
    e-mail

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