Abstract
In this paper, we are starting to construct a new theory of absolutely simple p-summing operators. We define a significant class of weak operator ideals, namely the class of absolutely simple p-summing operators between arbitrary real Banach spaces and show some basic properties of that class. A key feature of the resulting class is computing simple p-summing norms exactly for any linear operator between finite-dimensional normed spaces, in contrast to the computation of p-summing norms which is in general difficulty or the computation of Lipschitz p-summing norms between particular classes of metric spaces. Building upon S. Kwapień’s result, we figure out the relations between 2-summing norms and simple 2-summing norms and find out the relations between simple p-summing norms and diverse familiar norms of some linear operators. In the end, we present some concluding remarks and introduce some open problems that we think are intriguing.
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References
Achour, D., Dahia, E., Saleh, M.A.S.: Multilinear mixing operators and Lipschitz mixing operator ideals. Oper. Matrices 12, 903–931 (2018)
Botelho, G., Maia, M., Pellegrino, D., Santos, J.: A unified factorization theorem for Lipschitz summing operators. Q. J. Math. 70, 1521–1533 (2019)
Botelho, G., Pellegrino, D., Rueda, P.: A unified Pietsch Domination Theorem. J. Math. Anal. Appl. 365, 269–276 (2010)
Chávez-Domínguez, J.A.: Lipschitz \((q;p)\)-mixing operators. Proc. Am. Math. Soc. 140, 3101–3115 (2012)
Farmer, J.D., Johnson, W.B.: Lipschitz \(p\)-summing operators. Proc. Am. Math. Soc. 137, 2989–2995 (2009)
Kwapień, S.: Some Applications of the Theory of Absolutely Summing Operators. Aarhus Universitet, Matematisk Institut, Aarhus (1969)
Pellegrino, D., Santos, J.: A general Pietsch Domination Theorem. J. Math. Anal. Appl. 375, 371–374 (2011)
Pellegrino, D., Santos, J., Seoane-Sepúlveda, J.B.: Some techniques on nonlinear analysis and applications. Adv. Math. 229, 1235–1265 (2012)
Pietsch, A.: Absolut \(p\)-summierende Abbildungen in normierten Räumen. Stud. Math. 28, 333–353 (1967)
Pietsch, A.: Operator Ideals. Deutsch. Verlag Wiss., Berlin (1978). [North-Holland, Amsterdam–London–New York–Tokyo (1980)]
Saleh, M.A.S.: New types of Lipschitz summing maps between metric spaces. Math. Nachr. 290, 1347–1373 (2017)
Saleh, M.A.S.: Interpolative Lipschitz ideals. Colloq. Math. 163, 153–170 (2021)
Saleh, M.A.S.: Computations of Lipschitz summing norms and applications. Colloq. Math. 165, 31–40 (2021)
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We would like to express our gratitude to the referees for careful reading of the manuscript, valuable comments, and suggestions that led to improvement of the final version of the manuscript.
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Communicated by Dragan S. Djordjevic.
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Saleh, M.A.S., Shaakir, L.K. Absolutely simple p-summing operators and applications. Adv. Oper. Theory 9, 57 (2024). https://doi.org/10.1007/s43036-024-00356-3
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DOI: https://doi.org/10.1007/s43036-024-00356-3