Open Access

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


Abstract

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.


Full Text





Article Information

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


Google Scholar Metrics

Citing articles in Google Scholar
Similar articles in Google Scholar

Export article

Save to Mendeley



Citation Example

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.


BibTex

@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},
       URL = {http://mfat.imath.kiev.ua/article/?id=516},
}


All Issues