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# Wydawnictwa / Banach Center Publications / Wszystkie tomy

## Linear operators on non-locally convex Orlicz spaces

### Tom 79 / 2008

Banach Center Publications 79 (2008), 157-165 MSC: 46E30, 47B38. DOI: 10.4064/bc79-0-12

#### Streszczenie

We study linear operators from a non-locally convex Orlicz space $L^\Phi$ to a Banach space $(X,\|\cdot\|_X)$. Recall that a linear operator $T:L^\Phi\to X$ is said to be $\sigma$-smooth whenever $u_n\mathrel{\mathop{\longrightarrow}\limits^{{({\rm o})}}} 0$ in $L^\Phi$ implies $\|T(u_n)\|_X\to 0$. It is shown that every $\sigma$-smooth operator $T:L^\Phi\to X$ factors through the inclusion map $j:L^\Phi\to L^{{\overline{\Phi\hskip-.3pt}\hskip.3pt}}$, where ${\overline{\Phi\hskip-.3pt}\hskip.3pt}$ denotes the convex minorant of $\Phi$. We obtain the Bochner integral representation of $\sigma$-smooth operators $T:L^\Phi\to X$. This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on a locally convex Orlicz space.

#### Autorzy

• Marian NowakFaculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra
Szafrana 4a
65-516 Zielona Góra, Poland
e-mail
• Agnieszka OelkeFaculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra
Szafrana 4a
65-516 Zielona Góra, Poland
e-mail

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