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Weakly compact bilinear operators among real interpolation spaces

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We show a necessary and sufficient condition for weak compactness of bilinear operators interpolated by the real method. This characterization does not hold for interpolated operators by the complex method.
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[1] B. Beauzamy, Espaces d'Interpolation Réels: Topologie et Géométrie, Springer, Lecture Notes in Math. 666, Berlin, 1978. [2] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976. [3] Y. Brudny and N. Krugljak, Interpolation Functors and Interpolation Spaces, Vol. 1, North-Holland, Amsterdam, 1991. [4] A.P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964) 113-190. [5] F. Cobos, L.M. Fernández-Cabrera A. Manzano and A. Martínez, Real interpolation and closed operator ideals, J. Math. Pures el Appl. 83 (2004) 417-432. [6] F. Cobos, L.M. Fernández-Cabrera and A. Martínez, Interpolation of compact bilinear operators among quasi-Banach spaces and applications, Math. Nachr. 291 (2018) 2168-2187. [7] F. Cobos, L.M. Fernández-Cabrera and A. Martínez, On compactness results of Lions-Peetre type for bilinear operators, Nonlinear Anal. 199 (2020) 111951. [8] F. Cobos, L.M. Fernández-Cabrera and A. Martínez, On interpolation of weakly compact bilinear operators, Math. Nachr. 295 (2022) 1279-1291. [9] F. Cobos, T. Kühn and T. Schonbek, One-sided compactness results for Aronszajn-Gagliardo functors, J. Funct. Anal. 106 (1992) 274-313. [10] F. Cobos and A. Martínez, Remarks on interpolation properties of the measure of weak non-compactness and ideal variations, Math. Nachr. 208 (1999) 93-100. [11] F. Cobos and A. Martínez, Extreme estimates for interpolated operators by the real method, J. London Math. Soc. 60 (1999) 860-870. [12] M. Cwikel, Real and complex interpolation and extrapolation of compact operators, Duke Math. J. 65 (1992) 333-343. [13] A. Defand and K. Floret, Tensor Norms and Operator Ideals, North-Holland Mathematics Studies 176, Amsterdam, 1993. [14] J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge Univ. Press, Cambridge, 1995. [15] J. Diestel and J. J. Ulh Jr., Vector Measures, Amer. Math. Soc. Surveys No. 15, Providence, Rhode Island, 1977. [16] D.L. Fernandez and E.B. da Silva, Interpolation of bilinear operators and compactness, Nonlinear Anal. 73 (2010) 526-537. [17] L.M. Fernández-Cabrera and A. Martínez, On interpolation properties of compact bilinear operators, Math. Nachr. 290 (2017) 1663-1677. [18] L.M. Fernández-Cabrera and A. Martínez, Real interpolation of compact bilinear operators, J. Fourier Anal. Appl. 24 (2018) 1181-1203. [19] S. Heinrich, Closed operator ideals and interpolation, J. Funct. Anal. 35 (1980) 397-411. [20] J.-L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math. 19 (1964) 5-68. [21] L. Maligranda and A. Quevedo, Interpolation of weakly compact operators, Arch. Math. 55 (1990) 280-284. [22] A. Manzano, P. Rueda and E. A. Sánchez-Pérez, Closed injective ideals of multilinear operators, related measures and interpolation, Math. Nachr. 293 (2020) 510-532. [23] A. Manzano, P. Rueda and E. A. Sánchez-Pérez, Closed surjective ideals of multilinear operators and interpolation, Banach J. Math. Anal. 15:27 (2021). [24] M. Mastylo, Interpolation spaces not containing l1, J. Math. Pures et Appl. 68 (1989) 153-162. [25] M. Mastylo, On interpolation of weakly compact operators, Hokkaido Math. J. 22 (1993) 105-114. [26] M. Mastylo and E.B. Silva, Interpolation of compact bilinear operators, Bull. Math. Sci. 10 (2020) 2050002. [27] A. Pietsch, Operator Ideals, North-Holland, Amsterdam, 1980. [28] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
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