NORM-ATTAINING NUCLEAR OPERATORS SHELDON DANTAS, MINGU JUNG, ´ OSCAR ROLD ´ AN, AND ABRAHAM RUEDA ZOCA Abstract. Given two Banach spaces X and Y , we introduce and study a concept of norm-attainment in the space of nuclear operators N (X, Y ) and in the projective tensor product space X  ⊗ π Y . We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in N (X, Y ) and in X  ⊗ π Y is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces X and Y which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces X and Y failing the approximation property in such a way that the class of elements in X  ⊗ π Y which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper. 1. Introduction One of the most classical topics in the theory of Banach spaces is the study of norm- attaining functions. As a matter of fact, one of the most famous characterizations of reflexivity, due to R. James, is described in terms of linear functionals which attain their norms (see, for instance, [16, Corollary 3.56]). In the same direction, E. Bishop and R. Phelps proved that the set of all norm-attaining linear functionals is dense in X ∗ (see [5]). This motivated J. Lindenstrauss to study the analogous problem for bounded linear operators in his seminal paper [27], where it was obtained for the first time an example of a Banach space such that the Bishop-Phelps theorem is no longer true for this class of functions. Consequently, this opened the gate for a crucial and vast research on the topic during the past fifty years in many different directions. Indeed, just to name a few, J. Bourgain, R.E. Huff, J. Johnson, W. Schachermayer, J.J. Uhl, J. Wolfe, and V. Zizler continued the study about the set of all linear operators which attain their norms ([6, 20, 21, 35, 36, 37]); M. Acosta, R. Aron, F.J. Aguirre, Y.S. Choi, R. Pay´a ([1, 3, 10] tackled problems in the same line involving bilinear mappings; D. Garc´ ıa and M. Maestre 2010 Mathematics Subject Classification. Primary 46B04; Secondary 46B25, 46A32, 46B20. Key words and phrases. Bishop-Phelps theorem; norm-attaining operators; nuclear operators; tensor products. The first author was supported by the project OPVVV CAAS CZ.02.1.01/0.0/0.0/16 019/0000778 and by the Estonian Research Council grant PRG877. The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003857). The third author was supported by the Spanish Ministerio de Ciencia, Innovaci´ on y Universidades, grant FPU17/02023, and by the MINECO and FEDER project MTM2017-83262-C2-1-P. The fourth author was supported by MICINN (Spain) Grant PGC2018-093794- B-I00 (MCIU, AEI, FEDER, UE), by Junta de Andaluc´ ıa Grant A-FQM-484-UGR18 and by Junta de Andaluc´ ıa Grant FQM-0185. 1 arXiv:2006.09871v1 [math.FA] 17 Jun 2020