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Joint spreading models and uniform approximation of bounded operators

Volume 253 / 2020

S. A. Argyros, A. Georgiou, A.-R. Lagos, Pavlos Motakis Studia Mathematica 253 (2020), 57-107 MSC: Primary 46B03, 46B06, 46B25, 46B28, 46B45. DOI: 10.4064/sm181126-26-3 Published online: 8 January 2020

Abstract

We investigate the following property for Banach spaces. A Banach space $X$ satisfies the Uniform Approximation on Large Subspaces (UALS) if there exists $C \gt 0$ with the following property: for any $A\in \mathcal {L}(X)$ and convex compact subset $W$ of $\mathcal {L}(X)$ for which there exists $\varepsilon \gt 0$ such that for every $x\in X$ there exists $B\in W$ with $\|A(x)- B(x)\|\le \varepsilon \|x\|$, there exists a subspace $Y$ of $X$ of finite codimension and a $B\in W$ with $\|(A-B)|_Y\|_{\mathcal {L}(Y,X)}\leq C\varepsilon $. We prove that a class of separable Banach spaces including $\ell _p$ for $1\le p \lt \infty $, and $C(K)$ for $K$ countable and compact, satisfy the UALS. On the other hand, every $L_p[0,1]$, for $1\le p\le \infty $ and $p\neq 2$, fails the property and the same holds for $C(K)$ where $K$ is an uncountable metrizable compact space. Our sufficient conditions for UALS are based on joint spreading models, a multidimensional extension of the classical concept of spreading model, introduced and studied in the present paper.

Authors

  • S. A. ArgyrosDepartment of Mathematics
    Faculty of Applied Sciences
    National Technical University of Athens
    Zografou Campus
    157 80, Athens, Greece
    e-mail
  • A. GeorgiouDepartment of Mathematics
    Faculty of Applied Sciences
    National Technical University of Athens
    Zografou Campus
    157 80, Athens, Greece
    e-mail
  • A.-R. LagosSchool of Electrical and Computer Engineering
    National Technical University of Athens
    Zografou Campus
    157 80 Athens, Greece
    e-mail
  • Pavlos MotakisDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    Urbana, IL 61801, U.S.A.
    e-mail

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