×

Dominated multilinear operators defined on tensor products of Banach spaces. (English) Zbl 1490.46070

Summary: In this paper, we introduce and study new classes of dominated multilinear operators, which we call \((p;p_1,\dots,p_n;G_1,\dots,G_n)\)-dominated and \((\tilde{p};p_1,\dots,p_n;G_1,\dots,G_n)\)-dominated multilinear operators defined on the tensor product of Banach spaces. Some characterizations of this type of operators are given and we prove some important coincidence results. As an application, we characterize \((p;p_1,\dots,p_n)\)-dominated multilinear operators on \(\mathcal{C}(K,G)\) and \((p;p_1,\dots,p_n)\)-dominated multilinear operators in the sense of Dinculeanu on \(\mathcal{C}(K,G)\), where \(K\) is a compact Hausdorff space and \(G\) a Banach space. We also treat the connection between an operator \(T\) and its associated operators \(T^t,\tilde{T}\) and \(T^{\#}\) for certain classes.

MSC:

46M05 Tensor products in functional analysis
46G25 (Spaces of) multilinear mappings, polynomials
47H60 Multilinear and polynomial operators
47A80 Tensor products of linear operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Achour, D.; Mezrag, L., On the Cohen strongly \(p\)-summing multilinear operators, J. Math. Anal. Appl., 327, 550-563 (2007) · Zbl 1121.47013 · doi:10.1016/j.jmaa.2006.04.065
[2] Blasco, O.; Signes, T., Remarks on \((q, p, Y)\)-summing operators, Quaest. Math., 26, 97-103 (2003) · Zbl 1042.47012 · doi:10.2989/16073600309486047
[3] Botelho, G., Cotype and absolutely summing multilinear mappings and homogeneous polynomials, Proc. R. Ir. Acad. Sect. A, 97, 145-153 (1997) · Zbl 0903.46018
[4] Defant, A., Floret, K.: Tensor Norms and Operator Ideals, vol. 176. North-Holland Mathematics Studies. Amsterdam (1993) · Zbl 0774.46018
[5] Dinculeanu, N., Vector Measures (1967), Berlin: Veb Deutscher Verlag der Wissenschaften, Berlin · Zbl 0156.14902
[6] Elezović, N., \({\tilde{p}} \)-summing operators defined on tensor products of Banach spaces, Glas. Mat. Ser., III, 24, 327-338 (1989) · Zbl 0822.47024
[7] Kislyakov, SV, Absolutely summing operators on the disc algebra, Algebra i Analiz, 3, 1-77 (1991) · Zbl 0761.47006
[8] Matos, MC, On a question of Pietsch about Hilbert-Schmidt multilinear mappings, J. Math. Anal. Appl., 257, 343-355 (2001) · Zbl 0993.46027 · doi:10.1006/jmaa.2000.7351
[9] Matos, MC, On multilinear mappings of nuclear type, Rev. Mat. Complut., 6, 61-81 (1993) · Zbl 0807.46022 · doi:10.5209/rev_REMA.1993.v6.n1.17846
[10] Maurey, B.: Rappels sur les operateurs sommants et radonifiants. Seminaire Maurey-Schwartz 1973/1974, Exposé No I · Zbl 0297.47017
[11] Mezrag, L.; Saadi, K., Inclusion and coincidence properties for Cohen strongly summing multilinear operators, Collect. Math., 64, 395-408 (2013) · Zbl 1307.47068 · doi:10.1007/s13348-012-0071-2
[12] Montgomery-Smith, S.; Saab, P., \(p\)-summing operators on injective tensor product of spaces, Proc. R. Soc. Edinb. Sect. A, 120, 283-296 (1992) · Zbl 0793.47017 · doi:10.1017/S0308210500032145
[13] Pellegrino, D.; Santos, J.; Seoane-Sepúlveda, JB, Some techniques on nonlinear analysis and applications, Adv. Math., 229, 1235-1265 (2012) · Zbl 1248.47024 · doi:10.1016/j.aim.2011.09.014
[14] Pérez-García, D., Comparing different classes of absolutely summing multilinear operators, Arch. Math., 85, 258-267 (2005) · Zbl 1080.47047 · doi:10.1007/s00013-005-1125-4
[15] Popa, D., 2-summing multiplication operators, Stud. Math., 216, 77-96 (2013) · Zbl 1288.47022 · doi:10.4064/sm216-1-6
[16] Popa, D., \((r, p)\)-absolutely summing operators on the space \({\cal{C}}(T, X)\) and applications, Abstr. Appl. Anal., 6, 309-315 (2001) · Zbl 1035.47011 · doi:10.1155/S1085337501000434
[17] Popa, D., Dominated \(n\)-linear operators with respect to a system of \(n\) bounded linear operators, Linear Multilinear Algebra, 65, 1097-1107 (2017) · Zbl 1422.47069 · doi:10.1080/03081087.2016.1228814
[18] Popa, D., Remarks on multiple summing operators on \({\cal{C}}(\Omega )\)-spaces, Positivity, 18, 29-39 (2014) · Zbl 1315.47053 · doi:10.1007/s11117-013-0228-6
[19] Ryan, RA, Introduction to Tensor Products of Banach Spaces (2002), London: Springer, London · Zbl 1090.46001 · doi:10.1007/978-1-4471-3903-4
[20] Swartz, C., Absolutely summing and dominated operators on spaces of vector-valued continuous functions, Trans. Am. Math. Soc., 179, 123-131 (1973) · Zbl 0226.46038 · doi:10.1090/S0002-9947-1973-0320796-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.