Abstract
In this article, we study the ideals of mid p-summing operators and obtain a representation by tensor norms for these operator ideals. These tensor norms are defined using a particular kind of sequential dual of the class of mid p-summable sequences. As a consequence, we prove a characterization of the adjoints of weakly and absolutely mid p-summing operators in terms of the operators that are defined by the transformation of dual spaces of certain vector-valued sequence spaces. Further, we study the natural norms induced on \(\ell _{p}\otimes X\) from the class of mid p-summable sequences and its dual sequence class. These norms connect the ideals of mid p-summing operators and the associated tensor norms to the calculus of traced tensor norms which provides several interesting results about these ideals and the norms associated with them.
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References
Botelho, G., Campos, J.R.: On the transformation of vector-valued sequences by linear and multilinear operators. Mon. Math. 183, 415–435 (2017)
Botelho, G., Campos, J.R.: Duality theory for generalized summing linear operators. Collect. Math. 1–16 (2022)
Botelho, G., Campos, J.R., Santos, J.: Operator ideals related to absolutely summing and Cohen strongly summing operators. Pac. J. Math. 287, 1–17 (2017)
Botelho, G., Campos, J.R., Nascimento, L.: Maximal ideals of generalized summing linear operators. ar**v:2206.02944 (2022)
Chevet, S.: Sur certains produits tensoriels topologiques d’espaces de Banach. Z. Wahrscheinlichkeit. Verwandte Gebiete 11, 120–138 (1969)
Defant, A., Floret, K.: Tensor Norms and Operator Ideals. Elsevier, Amsterdam (1992)
Fourie, J., Zeekoi, E.D.: Classes of sequentially limited operators. Glasg. Math. J. 58, 573–586 (2016)
Grothendieck, A.: Résumé de la théorie métrique des produits tensoriels topologiques. Soc. Mat. S\({\tilde{a}}\)o Paulo (1956)
Karn, A., Sinha, D.: An operator summability of sequences in Banach spaces. Glasg. Math. J. 56, 427–437 (2014)
Pietsch, A.: Operator Ideals. Elsevier, Amsterdam (1980)
Ryan, R.: Introduction to Tensor Products of Banach Spaces. Springer, London (2013)
Saphar, P.: Applications á puissance nucléaire et applications de hilbert-schmidt dans les espaces de banach. C. R. Hebd. Seances Acad. Sci. 261, 867 (1965)
Saphar, P.: Produits tensoriels d’espaces de Banach et classes d’applications linéaires. Stud. Math. 38, 71–100 (1970)
Schatten, R.: A Theory of Cross-spaces. Princeton University Press, Princeton (1985)
Zeekoi, E.D.: Classes of Dunford-Pettis-type operators with applications to Banach spaces and Banach lattices. Ph.D. Thesis, North-West University, Potchefstroom Campus, South Africa (2017)
Acknowledgements
The authors would like to thank the anonymous referee for the suggestions and careful reading. The first author acknowledges the financial support from the Department of Science and Technology, India (Grant No. DST/INSPIRE Fellowship/2019/IF190043).
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Communicated by Miguel Martin.
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Philip, A., Baweja, D. Ideals of mid \({\varvec{p}}\)-summing operators: a tensor product approach. Adv. Oper. Theory 8, 28 (2023). https://doi.org/10.1007/s43036-023-00256-y
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DOI: https://doi.org/10.1007/s43036-023-00256-y