Abstract
We introduce and study the Lipschitz injective hull of Lipschitz operator ideals defined between metric spaces. We show some properties and apply the results to the ideal of Lipschitz p-nuclear operators, obtaining the ideal of Lipschitz quasi p-nuclear operators. Also, we introduce in a natural way the ideal of Lipschitz Pietsch p-integral operators and show that its Lipschitz injective hull coincide with the ideal of Lipschitz p-summing operators defined by Farmer and Johnson. Finally, we consider both ideals as Lipschitz operator ideals between a metric space and a Banach space, showing that these ideals are not of composition type. Their maximal hull and minimal kernel are also studied.
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Acknowledgements
We would like to thank to R. Villafañe for very helpful conversations. Also, we thank the referee for his/her comments and suggestions which helped us to improve this article substantially. D. Achour and E. Dahia acknowledges with thanks the support of the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria. P. Turco was supported in part by CONICET PIP 11220130100483, ANPCyT PICT-2015-2299.
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Communicated by Jari Taskinen.
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Achour, D., Dahia, E. & Turco, P. The Lipschitz injective hull of Lipschitz operator ideals and applications. Banach J. Math. Anal. 14, 1241–1257 (2020). https://doi.org/10.1007/s43037-020-00060-3
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DOI: https://doi.org/10.1007/s43037-020-00060-3