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Interpolation of bilinear operators and compactness. (English) Zbl 1200.46023

The authors analyse the behavior of bilinear operators acting on interpolation spaces generated by the general real method in relation to compactness. They prove analogs of classical compactness theorems (in the linear case) due to J.L.Lions and J.Peetre [Publ.Math., Inst.Hautes Étud.Sci.19, 5–68 (1964; Zbl 0148.11403)], K.Hayakawa [J. Math.Soc.Japan21, 189–199 (1969; Zbl 0181.13703)], and A.Persson [Ark.Mat.5, 215–219 (1964; Zbl 0128.35204)]. However, the authors neither mention that, due to M.Cwikel [Duke Math.J.65, 333–343 (1992; Zbl 0787.46062)], the linear case was solved in full generality, nor explain why they cannot provide such a general result for the bilinear case; recall that Cwikel deduced the full general result from Hayakawa’s result (where not only one, but both restrictions are assumed to be compact) by reiteration and dealing with \(K\)-functionals – it would have been nice to see why these techniques fail (?) when passing over to the bilinear case.

MSC:

46B70 Interpolation between normed linear spaces
46B50 Compactness in Banach (or normed) spaces
46M35 Abstract interpolation of topological vector spaces
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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References:

[1] Calderón, A. P., Intermediate spaces and interpolation, the complex method, Studia Math., 24, 113-190 (1964) · Zbl 0204.13703
[2] Janson, S., On interpolation of multi-linear operators, function spaces and applications, (Proc. US-Swed. Semin.. Proc. US-Swed. Semin., Lect. Notes Math., vol. 1302 (1988), Springer-Verlag: Springer-Verlag Lund, Sweden), 290-302 · Zbl 0827.46062
[3] Maligranda, L., Interpolation of some spaces of Orlicz type: II bilinear interpolation, Bull. Pol. Acad. Sci. Math., 37, 7-12, 453-457 (1989) · Zbl 0758.46043
[4] Mastylo, M., On interpolation of bilinear operators, J. Funct. Anal., 214, 2, 260-283 (2004) · Zbl 1065.46017
[5] Lions, J. L.; Peetre, J., Sur une classe d’espaces d’interpolation, Publ. Math. Inst. Hautes Études Sci., 19, 5-68 (1964) · Zbl 0148.11403
[6] Berg, J.; Löfströn, J., Interpolation Spaces. An Introduction (1976), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York
[7] Gustavsson, J., A function parameter in connection with interpolation of Banach spaces, Math. Scand., 42, 289-305 (1978) · Zbl 0389.46024
[8] Gustavsson, J.; Peetre, J., Interpolation of Orlicz spaces, Studia Math., 60, 33-59 (1977) · Zbl 0353.46019
[9] Boyd, D. W., Indices of function spaces and their relationship to interpolation, Canad. J. Math., 21, 1245-1254 (1969) · Zbl 0184.34802
[10] Boyd, D. W., Indices for the Orlicz spaces, Pacific J. Math., 38, 315-323 (1971) · Zbl 0227.46039
[11] Cobos, F.; Peetre, J., Interpolation of compactness using Aronszajn-Gagliardo functors, Israel J. Math., 68, 220-240 (1989) · Zbl 0716.46054
[12] Hayakawa, K., Interpolation by the real method preserves compactness of operators, J. Math. Soc. Japan, 21, 189-199 (1969) · Zbl 0181.13703
[13] Persson, A., Compact linear mappings between interpolations spaces, Ark. Mat., 5, 215-219 (1964) · Zbl 0128.35204
[14] Krasnoselskii, M. A., On a theorem of M. Riesz, Dokl. Akad. Nauk SSSR, 132, 510-513 (1960)
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