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Operators defined on $L_1$ which "nowhere" attain their norm


Let $E$ be either $\ell_1$ of $L_1$. We consider $E$-unattainable continuous linear operators $T$ from $L_1$ to a Banach space $Y$, i.e., those operators which do not attain their norms on any subspace of $L_1$ isometric to $E$. It is not hard to see that if $T: L_1 \to Y$ is $\ell_1$-unattainable then it is also $L_1$-unattainable. We find some equivalent conditions for an operator to be $\ell_1$-unattainable and construct two operators, first $\ell_1$-unattainable and second $L_1$-unattainable but not $\ell_1$-unattainable. Some open problems remain unsolved.

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TitleOperators defined on $L_1$ which "nowhere" attain their norm
SourceMethods Funct. Anal. Topology, Vol. 16 (2010), no. 1, 17-27
MathSciNet   MR2656128
CopyrightThe Author(s) 2010 (CC BY-SA)

Authors Information

I. V. Krasikova
Department of Mathematics, Zaporizhzhya National University, 2 Zhukovs'koho, Zapo\-rizhzhya, Ukraine

V. V. Mykhaylyuk
Department of Mathematics, Chernivtsi National University, 2 Kotsyubyns'koho, Chernivtsi, 58012, Ukraine

M. M. Popov
Departamento de Analisis Matematico, Facultad de Ciencias, Universidad de Granada

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I. V. Krasikova, V. V. Mykhaylyuk, and M. M. Popov, Operators defined on $L_1$ which "nowhere" attain their norm, Methods Funct. Anal. Topology 16 (2010), no. 1, 17-27.


@article {MFAT516,
    AUTHOR = {Krasikova, I. V. and Mykhaylyuk, V. V. and Popov, M. M.},
     TITLE = {Operators defined on $L_1$ which "nowhere" attain their norm},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {16},
      YEAR = {2010},
    NUMBER = {1},
     PAGES = {17-27},
      ISSN = {1029-3531},
  MRNUMBER = {MR2656128},
       URL = {},

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