zbMATH — the first resource for mathematics

Type and cotype of multilinear operators. (English) Zbl 1385.47026
In this paper, the authors introduce the notions of type $$(p_1,\dots,p_n)$$ and cotype $$q$$ for continuous $$n$$-linear operators, extending the classical notion. A finite rank $$n$$-linear operator has any proper cotype only when $$n = 1$$  or $$2$$. It is also shown that this type of operators form a Banach ideal.

MSC:
 47L22 Ideals of polynomials and of multilinear mappings in operator theory 46G25 (Spaces of) multilinear mappings, polynomials
Full Text:
References:
 [1] Aron, R; Berner, P, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France, 106, 3-24, (1978) · Zbl 0378.46043 [2] Blasco, O; Botelho, G; Pellegrino, D; Rueda, P, Summability of multilinear mappings: Littlewood, Orlicz and beyond, Monat. Math., 163, 131-147, (2011) · Zbl 1246.46045 [3] Blasco, O; Fourie, J; Schoeman, I, On operator valued sequences of multipliers and $$R$$-boundedness, J. Math. Anal. Appl., 328, 7-23, (2007) · Zbl 1117.47015 [4] Botelho, G.: Ideals of polynomials generated by weakly compact operators. Note Mat. 25, 69-102 (2005/2006) · Zbl 1223.46047 [5] Botelho, G., Campos, J.R.: On the transformation of vector-valued sequences by linear and multilinear operators. arXiv:1410.4261 [math.FA] (2015) · Zbl 1434.46026 [6] Botelho, G; Galindo, P; Pellegrini, L, Uniform approximation on ideals of multilinear mappings, Math. Scand., 106, 301-319, (2010) · Zbl 1193.47061 [7] Botelho, G; Pellegrino, D; Rueda, P, On composition ideals of multilinear mappings and homogeneous polynomials, Publ. Res. Inst. Math. Sci., 43, 1139-1155, (2007) · Zbl 1169.46023 [8] Castillo, J; García, R; Gonzalo, R, Banach spaces in which all multilinear forms are weakly sequentially continuous, Studia Math., 136, 121-145, (1999) · Zbl 0948.46010 [9] Davie, AM; Gamelin, TW, A theorem on polynomial-star approximation, Proc. Am. Math. Soc., 106, 351-356, (1989) · Zbl 0683.46037 [10] Dimant, V; Gonzalo, R; Jaramillo, JA, Asymptotic structure, $$ℓ _p$$-estimates of sequences, and compactness of multilinear mappings, J. Math. Anal. Appl., 350, 680-693, (2009) · Zbl 1165.46006 [11] Defant, A., Floret, K.: Tensor norms and operator ideals, North-Holland Mathematics Studies 176, North-Hollland (1993) · Zbl 0774.46018 [12] Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. Cambridge University Press, Cambridge (1995) · Zbl 0855.47016 [13] Diestel, J., Jarchow, H. and Pietsch, A.: Operator ideals. In: Johnson, W., Lindenstrauss J. (eds.) Handbook of the geometry of Banach spaces, Vol. I, North-Holland (2001), pp. 437-496 · Zbl 1012.47001 [14] Dineen, S.: Complex analysis on infinite dimensional spaces. Springer, New York (1999) · Zbl 1034.46504 [15] Floret, K; Hunfeld, S, Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces, Proc. Am. Math. Soc., 130, 1425-1435, (2002) · Zbl 1027.46054 [16] Galicer, D; Villafañe, R, Coincidence of extendible vector-valued ideals with their minimal kernel, J. Math. Anal. Appl., 421, 1743-1766, (2015) · Zbl 1318.47079 [17] Gonzalo, R; Jaramillo, JA, Smoothness and estimates of sequences in Banach spaces, Israel J. Math., 89, 321-341, (1995) · Zbl 0823.46013 [18] Knaust, H., Odell, E.: Weakly null sequences with upper $$ℓ _p$$-estimates, Lecture Notes in Math. 1470, Springer, New York (1991) · Zbl 0759.46013 [19] Pellegrino, D; Santos, J, Absolutely summing multilinear operators: a panorama, Quaest. Math., 34, 447-478, (2011) · Zbl 1274.47001 [20] Pérez-García, D, Comparing different classes of absolutely summing multilinear operators, Arch. Math., 85, 258-267, (2005) · Zbl 1080.47047 [21] Pietsch, A.: Operator ideals. Springer, New York (1980) · Zbl 0434.47030 [22] Pietsch, A.: Ideals of multilinear functionals. In: Proceedings of the second international conference on operator algebras, ideals and their applications in theoretical physics, pp. 185-199, Teubner-Texte, Leipzig (1983) · Zbl 1246.46045 [23] Pisier, G.: Factorization of linear operators and geometry of Banach spaces. Am. Math. Soc. Providence R.I., CBMS 60 (1986) · Zbl 0588.46010
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.