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(Strongly-)Dunford–Pettis Operators and Narrow Operators

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Abstract

Let \({{\mathcal {M}}}\) be a semifinite von Neumann algebra. We show that an operator T from the predual \(L_1({{\mathcal {M}}},\tau )\) of \({{\mathcal {M}}}\) into a Banach space X is strongly Dunford–Pettis if and only if \(T\circ i: L_1({{\mathcal {M}}},\tau ) \cap {{\mathcal {M}}}\rightarrow _i L_1({{\mathcal {M}}},\tau ) \rightarrow _T X\) is compact. We also show that for a finite measure space \((A,\Sigma ,\nu )\) and a reflexive Banach space X, a linear bounded operator \(T:L_{1}(\nu ,X)\rightarrow c_0\) is a Dunford–Pettis operator if and only if T is a dominated operator. We also prove that all bounded operators T from \(L_p(0,1)\) into the Schatten–von Neumann r-class \( C_r\) are necessarily narrow whenever \(1\le p<2\), \(1\le r\le p\), which answers a question raised in Huang et al. (Mediterr. J Math 19, 2022). Finally, we show that for a reflexive Banach space X all Dunford–Pettis operators from \(T:L_{1}(\nu ,X)\rightarrow c_0\) are narrow.

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Date availability is not applicable to this article as no new date were created or analyzed in this study.

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Acknowledgements

The authors would like to thank the anonymous reviewer for helpful comments, which significantly improves the presentation of the paper.

Funding

J. Huang was supported by the NNSF of China (No. 12031004 and 12301160). M. Pliev was supported by the Ministry of Science and Education of Russian Federation (Grant Number 075-02-2023-939). F. Sukochev’s research was supported by the Australian Research Council (FL170100052).

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Authors and Affiliations

  1. Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Jinghao Huang

  2. North-Ossetian State University, Vladikavkaz, Russia, 362025

    Marat Pliev & Fedor Sukochev

  3. North Caucasus Center for Mathematical Research of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz, Russia, 362027

    Marat Pliev

  4. School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, Australia

    Fedor Sukochev

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  1. Jinghao Huang
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Correspondence to Jinghao Huang.

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Huang, J., Pliev, M. & Sukochev, F. (Strongly-)Dunford–Pettis Operators and Narrow Operators. Integr. Equ. Oper. Theory 95, 22 (2023). https://doi.org/10.1007/s00020-023-02739-2

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