RT Journal Article T1 On a problem of Lions concerning real interpolation spaces. The quasi-Banach case A1 Cobos, Fernando A1 Cwikel, M. A1 Kühn, Thomas AB We prove that, under a mild condition on a couple (A0;A1) of quasi-Banach spaces, all real interpolation spaces (A0;A1)θ,p with 0 < θ < 1 and 0 < p ≤ ∞ are different from each other. In the Banach case and for 1 ≤ p ≤ ∞ this was shown by Janson, Nilsson, Peetre and Zafran, thus solving an old problem posed by J.-L. Lions. Moreover, we give an application to certain spaces which are important objects in Operator Theory and which consist of bounded linear operators whose approximation numbers belong to Lorentz sequence spaces. PB Elsevier SN 0022-247X YR 2022 FD 2022-08-28 LK https://hdl.handle.net/20.500.14352/71924 UL https://hdl.handle.net/20.500.14352/71924 LA eng NO [1] J.M. Almira and P. Fernández-Martínez, On slow decay of Peetre's K-functional. J. Math. Anal. Appl. 494 (2021) 124653.[2] C. Bennett and R. Sharpley, Interpolation of operators. Pure and Applied Mathematics, 129. Academic Press, Boston, MA, 1988.[3] J. Bergh and J. Löfström, Interpolation spaces. An introduction. Grundlehren der mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.[4] Yu. A. Brudnyi and N. Ya. Krugljak, Interpolation functors and interpolation spaces. North-Holland Mathematical Library, 47. North-Holland, Amsterdam, 1991.[5] A. Dvoretzky, Some results on convex bodies and Banach spaces. Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), 123-160, Jerusalem Academic Press, Jerusalem, and Pergamon, Oxford, 1961.[6] T. Figiel, A short proof of Dvoretzky's theorem on almost spherical sections of convex bodies. Compositio Math. 33 (1976), 297-301.[7] E. Gagliardo, A uni�ed structure in various families of function spaces. Compactness and closure theorems. Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), 237-241, Jerusalem Academic Press, Jerusalem, and Pergamon, Oxford, 1961.[8] S. Janson, P. Nilsson, J. Peetre and M. Zafran, Notes on Wolff's notes on interpolation spaces. Proc. London Math. Soc. (3), 48 (1984), 283-299.[9] H. König, Interpolation of operator ideals with an application to eigenvalue distribution problems. Math. Ann. 233 (1978), 35-48.[10] H. König, Eigenvalue distribution of compact operators. Operator Theory: Advances and Applications, Vol. 16, Birkhäuser, Basel, 1986.[11] G. Köthe, Topological vector spaces I. Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, 1969.[12] P. Krée, Interpolation d'espaces qui ne sont ni norms ni complets. Applications. Ann. Inst. Fourier 17 (1967), 137-174.[13] M. Lévy, L'espace d'interpolation réel (A0;A1)θ,p contient lp. C. R. Acad. Sci. Paris 289 (1979), 675-677.[14] M. Lévy, Structure �fine des espaces d'interpolation réels; application aux espaces de Lorentz, PhD thesis, Univ. Pierre et Marie Curie Paris, 1980.[15] J.A. López Molina, On subspaces isomorphic to lp in interpolation of quasi Banach spaces. Turkish J. Math. 24 (2000), 373-378.[16] J.A. López Molina, Existence of complemented subspaces isomorphic to lp in quasi Banach interpolation spaces. Rocky Mountain J. Math. 39 (2009), 899-926.[17] V. D. Milman, New proof of the theorem of A. Dvoretzky on intersections of convex bodies. (in Russian) Funktsional. Anal. i Prilozhen. 5 (1971), No. 4, 28-37.[18] J. Peetre and G. Sparr, Interpolation of normed Abelian groups. Ann. Mat. Pura Appl. 92 (1972), 217-262.[19] A. Pietsch, Operator ideals. VEB Deutscher Verlag der Wissenschaften, Berlin, 1978, and North-Holland Mathematical Library, 20. North-Holland Publishing Co., Amsterdam-New York, 1980.[20] A. Pietsch, Eigenvalues and s-numbers. Cambridge Studies in Advanced Mathematics, 13. Cambridge University Press, Cambridge, 1987.[21] G. Pisier, The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989.[22] W. Rudin, Functional analysis. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.[23] J. D. Stafney, Analytic interpolation of certain multiplier spaces. Pacifi�c J. Math. 32 (1970), 241-248.[24] A. Szankowski, On Dvoretzky's theorem on almost sperical sections of convex bodies. Israel J. Math. 17 (1974), 325-338.[25] H. Triebel, Über die Verteilung der Approximationszahlen kompakter Operatoren in Sobolev-Besov-Räumen. Invent. Math. 4 (1967), 275-293.[26] H. Triebel, Interpolation theory, function spaces, differential operators. VEB Deutscher Verlag der Wissenschaften, Berlin, 1978, and North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978 NO CRUE-CSIC (Acuerdos Transformativos 2022) NO Ministerio de Ciencia e Innovación (MICINN) DS Docta Complutense RD 1 dic 2023