Linear operators on non-locally convex Orlicz spaces

Marian Nowak; Agnieszka Oelke

Banach Center Publications (2008)

  • Volume: 79, Issue: 1, page 157-165
  • ISSN: 0137-6934

Abstract

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We study linear operators from a non-locally convex Orlicz space L Φ to a Banach space ( X , | | · | | X ) . Recall that a linear operator T : L Φ X is said to be σ-smooth whenever u ( o ) 0 in L Φ implies | | T ( u ) | | X 0 . It is shown that every σ-smooth operator T : L Φ X factors through the inclusion map j : L Φ L Φ ̅ , where Φ̅ denotes the convex minorant of Φ. We obtain the Bochner integral representation of σ-smooth operators T : L Φ X . This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on a locally convex Orlicz space.

How to cite

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Marian Nowak, and Agnieszka Oelke. "Linear operators on non-locally convex Orlicz spaces." Banach Center Publications 79.1 (2008): 157-165. <http://eudml.org/doc/282076>.

@article{MarianNowak2008,
abstract = {We study linear operators from a non-locally convex Orlicz space $L^Φ$ to a Banach space $(X,||·||_X)$. Recall that a linear operator $T:L^Φ → X$ is said to be σ-smooth whenever $uₙ\longrightarrow \limits ^\{(o)\} 0$ in $L^Φ$ implies $||T(uₙ)||_X → 0$. It is shown that every σ-smooth operator $T:L^Φ → X$ factors through the inclusion map $j:L^Φ → L^\{Φ̅\}$, where Φ̅ denotes the convex minorant of Φ. We obtain the Bochner integral representation of σ-smooth operators $T:L^Φ → X$. This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on a locally convex Orlicz space.},
author = {Marian Nowak, Agnieszka Oelke},
journal = {Banach Center Publications},
keywords = {Orlicz spaces; Lebesgue topology; modular topology; -smooth operators; Bochner integral representation; Radon-Nikodym property},
language = {eng},
number = {1},
pages = {157-165},
title = {Linear operators on non-locally convex Orlicz spaces},
url = {http://eudml.org/doc/282076},
volume = {79},
year = {2008},
}

TY - JOUR
AU - Marian Nowak
AU - Agnieszka Oelke
TI - Linear operators on non-locally convex Orlicz spaces
JO - Banach Center Publications
PY - 2008
VL - 79
IS - 1
SP - 157
EP - 165
AB - We study linear operators from a non-locally convex Orlicz space $L^Φ$ to a Banach space $(X,||·||_X)$. Recall that a linear operator $T:L^Φ → X$ is said to be σ-smooth whenever $uₙ\longrightarrow \limits ^{(o)} 0$ in $L^Φ$ implies $||T(uₙ)||_X → 0$. It is shown that every σ-smooth operator $T:L^Φ → X$ factors through the inclusion map $j:L^Φ → L^{Φ̅}$, where Φ̅ denotes the convex minorant of Φ. We obtain the Bochner integral representation of σ-smooth operators $T:L^Φ → X$. This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on a locally convex Orlicz space.
LA - eng
KW - Orlicz spaces; Lebesgue topology; modular topology; -smooth operators; Bochner integral representation; Radon-Nikodym property
UR - http://eudml.org/doc/282076
ER -

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