Some properties of 𝑝-limited sets

Galindo, Pablo; Miranda, Vinícius

doi:10.1090/proc/16573

Some properties of $p$-limited sets
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by Pablo Galindo and Vinícius C. C. Miranda;

Proc. Amer. Math. Soc. 152 (2024), 749-763

DOI: https://doi.org/10.1090/proc/16573

Published electronically: November 29, 2023

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Abstract:

Karn and Sinha [Glasg. Math. J. 56 (2014), pp. 427–437] introduced the $p$-limited ($1 \leq p < \infty$) sets (see the definition below). We show that $p$-limited sets are preserved by continuous polynomials as well as by the projective tensor product and that scalar-valued polynomials are $p$-summable on $p$-limited sets. Considering the notion of $p$-limited set from the $\ell _p$-valued operators point of view, we introduce in Section \ref{sec:3} two weaker types of $p$-limitedness in the setting of Banach lattices and study their basic properties.

References

Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Springer, Dordrecht, 2006. Reprint of the 1985 original. MR 2262133, DOI 10.1007/978-1-4020-5008-4
Richard M. Aron and Pilar Rueda, $p$-compact homogeneous polynomials from an ideal point of view, Function spaces in modern analysis, Contemp. Math., vol. 547, Amer. Math. Soc., Providence, RI, 2011, pp. 61–71. MR 2856480, DOI 10.1090/conm/547/10807
Richard M. Aron and Pilar Rueda, Ideals of homogeneous polynomials, Publ. Res. Inst. Math. Sci. 48 (2012), no. 4, 957–969. MR 2999547, DOI 10.2977/PRIMS/93
Jin Xi Chen, Zi Li Chen, and Guo Xing Ji, Almost limited sets in Banach lattices, J. Math. Anal. Appl. 412 (2014), no. 1, 547–553. MR 3145821, DOI 10.1016/j.jmaa.2013.10.085
Juan Manuel Delgado and Cándido Piñeiro, On $p$-compact sets in classical Banach spaces, Int. Math. Forum 9 (2014), no. 1-4, 51–63. MR 3173805, DOI 10.12988/imf.2014.311215
J. M. Delgado and C. Piñeiro, A note on $p$-limited sets, J. Math. Anal. Appl. 410 (2014), no. 2, 713–718. MR 3111861, DOI 10.1016/j.jmaa.2013.08.045
Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR 1342297, DOI 10.1017/CBO9780511526138
Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, and Václav Zizler, Banach space theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. The basis for linear and nonlinear analysis. MR 2766381, DOI 10.1007/978-1-4419-7515-7
Pablo Galindo, Polynomials and limited sets, Proc. Amer. Math. Soc. 124 (1996), no. 5, 1481–1488. MR 1307518, DOI 10.1090/S0002-9939-96-03175-9
Pablo Galindo and Vinícius C. C. Miranda, Grothendieck-type subsets of Banach lattices, J. Math. Anal. Appl. 506 (2022), no. 1, Paper No. 125570, 14. MR 4304813, DOI 10.1016/j.jmaa.2021.125570
Pablo Galindo and V. C. C. Miranda, A class of sets in a Banach space coarser than limited sets, Bull. Braz. Math. Soc. (N.S.) 53 (2022), no. 3, 941–955. MR 4458795, DOI 10.1007/s00574-022-00290-z
Ioana Ghenciu, A note on $p$-limited sets in dual Banach spaces, Monatsh. Math. 200 (2023), no. 2, 255–270. MR 4544297, DOI 10.1007/s00605-022-01738-6
Anil Kumar Karn and Deba Prasad Sinha, An operator summability of sequences in Banach spaces, Glasg. Math. J. 56 (2014), no. 2, 427–437. MR 3187909, DOI 10.1017/S0017089513000360
M. L. Lourenço and V. C. C. Miranda, The property (d) and the almost limited completely continuous operators, Anal. Math. 49 (2023), no. 1, 225–241. MR 4554794, DOI 10.1007/s10476-023-0190-x
Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093, DOI 10.1007/978-3-642-76724-1
Raymond A. Ryan, Weakly compact holomorphic mappings on Banach spaces, Pacific J. Math. 131 (1988), no. 1, 179–190. MR 917872, DOI 10.2140/pjm.1988.131.179
Deba P. Sinha and Anil K. Karn, Compact operators whose adjoints factor through subspaces of $l_p$, Studia Math. 150 (2002), no. 1, 17–33. MR 1893422, DOI 10.4064/sm150-1-3
Elroy D. Zeekoei and Jan H. Fourie, On $p$-convergent operators on Banach lattices, Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 5, 873–890. MR 3785686, DOI 10.1007/s10114-017-7172-5