Dieudonné operators on the space of Bochner integrable functions

Volume 92 / 2011

Marian Nowak Banach Center Publications 92 (2011), 279-282 MSC: 47B38, 47B05, 46E40. DOI: 10.4064/bc92-0-19


A bounded linear operator between Banach spaces is called a Dieudonné operator (=weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $X$ and $Y$ be Banach spaces. We study Dieudonné operators $T:L^1(X)\to Y$. Let $i_\infty:L^\infty(X) \to L^1(X)$ stand for the canonical injection. We show that if $X$ is almost reflexive and $T:L^1(X)\to Y$ is a Dieudonné operator, then $T\circ i_\infty:L^\infty(X)\to Y$ is a weakly compact operator. Moreover, we obtain that if $T:L^1(X)\to Y$ is a bounded linear operator and $T\circ i_\infty:L^\infty(X)\to Y$ is weakly compact, then $T$ is a Dieudonné operator.


  • Marian NowakFaculty of Mathematics, Computer Science and Econometrics
    University of Zielona Góra
    ul. Prof. Szafrana 4a
    65-516 Zielona Góra, Poland

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