×

Operator \(p\)-compact mappings. (English) Zbl 1430.46041

In 2002, inspired by Grothendieck’s classical result that characterizes the compact sets in Banach spaces as those sets that lie in the absolutely convex hull of a null sequence, in [D. P. Sinha and A. K. Karn, Stud. Math 150, No. 1, 17–33 (2002; Zbl 1008.46008)], the concept of \(p\)-compact sets in Banach spaces, \(1\leq p < \infty\), is formalized. These are, loosely speaking, those compact sets that are determined by \(p\)-summable sequences. Related to this, the concept of \(p\)-compact linear operators arises naturally. These are the operators that map bounded sets into \(p\)-compact sets. The \(p\)-compact operators have been studied in the last 10 years and several properties are well known. For example, the ideal of \(p\)-compact operators coincides with the surjective hull of the ideal of the right \(p\)-nuclear operators [J. M. Delgado et al., Stud. Math 197, No. 3, 291–304 (2010; Zbl 1190.47024)]). Also, the \(p\)-compact operators have a special factorization through a quotient of \(\ell_{p'}\).
In this article, the authors extend the notion of \(p\)-compactness into the operator space setting. The way they do it is extending the factorization of \(p\)-compact linear operators. For this, the authors take from J. A. Chávez-Domínguez [Houston J. Math. 42, No. 2, 577–596 (2016; Zbl 1358.46056)] the notion of \(p\)-right Chevet-Saphar operator space tensor norm and introduce and study the completely right \(p\)-nuclear mappings between operator spaces. They get a factorization of this class and then they use it to define the operator \(p\)-compact mappings.
Some properties of the \(p\)-compact linear operators are extended. For example, the ideal of operator \(p\)-compact mappings coincides with the surjective hull of the completely right \(p\)-nuclear maps. In addition, the authors obtain a characterization of the operator \(p\)-compact mappings that can be compared with that of the operator compact mapping, introduced by Webster in his doctoral Thesis. This characterization shows, in some sense, that the operator compact mappings of Webster can be considered as the class of operator \(\infty\)-compact maps.
The article finishes with some questions regarding the tensor norm associated to the mapping ideal of operator \(p\)-compact mappings.

MSC:

46L07 Operator spaces and completely bounded maps
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46B28 Spaces of operators; tensor products; approximation properties
47L25 Operator spaces (= matricially normed spaces)
47B07 Linear operators defined by compactness properties
47L20 Operator ideals
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aron, R.; Çalışkan, E.; García, D.; Maestre, M., Behavior of holomorphic mappings on \(p\)-compact sets in a Banach space, Trans. Amer. Math. Soc., 368, 7, 4855-4871 (2016) · Zbl 1346.46039
[2] Blecher, D. P., Tensor products of operator spaces. II, Canad. J. Math., 44, 1, 75-90 (1992) · Zbl 0787.46059
[3] Blecher, D. P.; Le Merdy, C., Operator Algebras and Their Modules—An Operator Space Approach, London Mathematical Society Monographs. New Series, vol. 30 (2004), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press Oxford, Oxford Science Publications · Zbl 1061.47002
[4] Blecher, D. P.; Paulsen, V. I., Tensor products of operator spaces, J. Funct. Anal., 99, 2, 262-292 (1991) · Zbl 0786.46056
[5] Chávez-Domínguez, J. A., The Chevet-Saphar tensor norms for operator spaces, Houston J. Math., 42, 2, 577-596 (2016) · Zbl 1358.46056
[6] J.A. Chávez-Domínguez, V. Dimant, D. Galicer, Operator space tensor norms, in preparation.; J.A. Chávez-Domínguez, V. Dimant, D. Galicer, Operator space tensor norms, in preparation.
[7] Choi, Y. S.; Kim, J. M., The dual space of \((L(X, Y), \tau_p)\) and the \(p\)-approximation property, J. Funct. Anal., 259, 9, 2437-2454 (2010) · Zbl 1211.46014
[8] Defant, A.; Floret, K., Tensor Norms and Operator Ideals, North-Holland Mathematics Studies, vol. 176 (1993), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0774.46018
[9] Delgado, J. M.; Pineiro, C.; Serrano, E., Operators whose adjoints are quasi \(p\)-nuclear, Studia Math., 197, 291-304 (2010) · Zbl 1190.47024
[10] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, vol. 43 (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0855.47016
[11] Dimant, V.; Fernández-Unzueta, M., Biduals of tensor products in operator spaces, Studia Math., 230, 2, 165-185 (2015) · Zbl 1353.47116
[12] Effros, E. G.; Ruan, Z.-J., Operator Spaces, London Mathematical Society Monographs. New Series, vol. 23 (2000), The Clarendon Press Oxford University Press: The Clarendon Press Oxford University Press New York
[13] Effros, E. G.; Junge, M.; Ruan, Z.-J., Integral mappings and the principle of local reflexivity for noncommutative \(L^1\)-spaces, Ann. of Math. (2), 151, 1, 59-92 (2000) · Zbl 0957.47051
[14] Fourie, J. H., Injective and surjective hulls of classical \(p\)-compact operators with application to unconditionally \(p\)-compact operators, Studia Math., 240, 147-159 (2018) · Zbl 1464.47015
[15] Fourie, J. H.; Swart, J., Banach ideals of \(p\)-compact operators, Manuscripta Math., 26, 4, 349-362 (1979) · Zbl 0403.47012
[16] Galicer, D.; Lassalle, S.; Turco, P., The ideal of \(p\)-compact operators: a tensor product approach, Studia Math., 211, 269-286 (2012) · Zbl 1269.47052
[17] Grothendieck, A., Produits tensoriels topologiques et espaces nucléaires, vol. 16 (1955), American Mathematical Soc. · Zbl 0064.35501
[18] Grothendieck, A., Résumé de la théorie métrique des produits tensoriels topologiques (1956), Soc. de Matemática de São Paulo · Zbl 0074.32303
[19] Junge, M., Factorization Theory for Spaces of Operators (1996), Habilitation Thesis, Kiel
[20] Junge, M., Factorization Theory for Spaces of Operators (1999), Institut for Matematik og Datalogi Odense Universitet
[21] Junge, M.; Parcet, J., Maurey’s factorization theory for operator spaces, Math. Ann., 347, 2, 299-338 (2010) · Zbl 1213.46049
[22] Lassalle, S.; Turco, P., The Banach ideal of \(A\)-compact operators and related approximation properties, J. Funct. Anal., 265, 10, 2452-2464 (2013) · Zbl 1298.47031
[23] Lindenstrauss, J.; Pełczyński, A., Absolutely summing operators in \(L_p\)-spaces and their applications, Studia Math., 29, 3, 275-326 (1968) · Zbl 0183.40501
[24] Muñoz, F.; Oja, E.; Piñeiro, C., On \(α\)-nuclear operators with applications to vector-valued function spaces, J. Funct. Anal., 269, 9, 2871-2889 (2015) · Zbl 1332.46028
[25] Oikhberg, T., Completely bounded and ideal norms of multiplication operators and Schur multipliers, Integral Equations Operator Theory, 66, 3, 425-440 (2010) · Zbl 1233.47033
[26] Oja, E., Grothendieck’s nuclear operator theorem revisited with an application to \(p\)-null sequences, J. Funct. Anal., 263, 9, 2876-2892 (2012) · Zbl 1301.47030
[27] Oja, E., A remark on the approximation of \(p\)-compact operators by finite-rank operators, J. Math. Anal. Appl., 387, 2, 949-952 (2012) · Zbl 1241.46013
[28] Pietsch, A., Ideale von operatoren in Banachräumen, Mitt. Math. Gesselsch. DDR, 1, 1-14 (1968) · Zbl 0174.44202
[29] Pietsch, A., Operator Ideals, vol. 16 (1978), Deutscher Verlag der Wissenschaften · Zbl 0399.47039
[30] Pietsch, A., Operator Ideals, North-Holland Mathematical Library, vol. 20 (1980), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam, translated from German by the author · Zbl 0399.47039
[31] Pietsch, A., The ideal of \(p\)-compact operators and its maximal hull, Proc. Amer. Math. Soc., 142, 2, 519-530 (2014) · Zbl 1293.47019
[32] Pisier, G., The operator Hilbert space OH, complex interpolation and tensor norms, Mem. Amer. Math. Soc., 122, 585 (1996), viii+103 · Zbl 0932.46046
[33] Pisier, G., Non-commutative vector valued \(L_p\)-spaces and completely \(p\)-summing maps, Astérisque, 247 (1998), vi+131 · Zbl 0937.46056
[34] Pisier, G., Introduction to Operator Space Theory, London Mathematical Society Lecture Note Series, vol. 294 (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1093.46001
[35] Reinov, O., On linear operators with \(p\)-nuclear adjoints, Vestn. SPbGU, Mat., 4, 24-27 (2000) · Zbl 1044.47508
[36] Ryan, R. A., Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics (2002), Springer-Verlag London Ltd.: Springer-Verlag London Ltd. London · Zbl 1090.46001
[37] Sinha, D. P.; Karn, A. K., Compact operators whose adjoints factor through subspaces of lp, Studia Math., 150, 1, 17-33 (2002) · Zbl 1008.46008
[38] Webster, C., Local Operator Spaces and Applications (1997), University of California, PhD thesis
[39] Webster, C., Matrix compact sets and operator approximation properties (1998)
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.