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Complex interpolation and regular operators between Banach lattices. (English) Zbl 0991.46007

Summary: We study certain interpolation and extension properties of the space of regular operators between two Banach lattices. Let \(R_p\) be the space of all the regular (or equivalently order bounded) operators on \(L_p\) equipped with the regular norm. We prove the isometric identity \(R_p = (R_\infty,R_1)^\theta\) if \(\theta = 1/p\), which shows that the spaces \((R_p)\) form an interpolation scale relative to Calderón’s interpolation method. We also prove that if \(S\subset L_p\) is a subspace, every regular operator \(u : S \to L_p\) admits a regular extension \(\widetilde u : L_p \to L_p\) with the same regular norm. This extends a result due to Mireille Lévy in the case \(p = 1\). Finally, we apply these ideas to the Hardy space \(H^p\) viewed as a subspace of \(L_p\) on the circle. We show that the space of regular operators from \(H^p\) to \(L_p\) possesses a similar interpolation property as the spaces \(R_p\) defined above.

MSC:

46B70 Interpolation between normed linear spaces
46M35 Abstract interpolation of topological vector spaces
46B42 Banach lattices
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[1] J. Bergh, On the relation between the two complex methods of interpolation. Indiana Univ. Math. J.28, 775-777 (1979). · Zbl 0431.41003 · doi:10.1512/iumj.1979.28.28054
[2] J.Bergh and J.L?fstr?m, Interpolation spaces. An introduction. Berlin-Heidelberg-New York 1976.
[3] O. Blasco andQ. Xu, Interpolation between vector valued Hardy spaces. J. Funct. Anal.102, 331-359 (1991). · Zbl 0759.46066 · doi:10.1016/0022-1236(91)90125-O
[4] A. Calder?n, Intermediate spaces and interpolation, the complex method. Studia Math.24, 113-190 (1964). · Zbl 0204.13703
[5] U. Haagerup andG. Pisier, Factorization of analytic functions with values in noncommutativeL 1 spaces and applications. Canad. J. Math.41, 882-906 (1989). · Zbl 0821.46074 · doi:10.4153/CJM-1989-041-6
[6] A.Hess and G.Pisier, On theK t ,-functional for the coupleB(L 1,L 1),B(L ?,L in?)). Quart. J. Math. Oxford Ser. (2). Submitted.
[7] J. L.Krivine, Th?or?mes de factorisation dans les espaces de Banach r?ticul?s. S?minaire Maurey-Schwartz 73/74, Expos? 22, Ecole Polytechnique, Paris.
[8] M.L?vy, Prolongement d’un op?rateur d’un sous-espace deL 1(?) dansL 1(?). S?minaire d’Analyse Fonctionnelle 1979-1980. Expos? 5. Ecole Polytechnique, Palaiseau.
[9] M.Ledoux and M.Talagrand, Probability in Banach spaces. Berlin-Heidelberg-New York 1991. · Zbl 0748.60004
[10] J.Lindenstrauss and L.Tzafriri, Classical Banach spaces II, Function spaces. Berlin-Heidelberg-New York 1979. · Zbl 0403.46022
[11] P.Meyer-Nieberg, Banach Lattices, Berlin-Heidelberg-New York 1991.
[12] G. Pisier, Interpolation ofH p -spaces and noncommutative generalizations I. Pacific J. Math.155, 341-368 (1992). · Zbl 0747.46050
[13] G. Pisier, Interpolation ofH p -spaces and noncommutative generalizations II. Rev. Mat. Iberoamericana9, 281-291 (1993). · Zbl 0788.46071
[14] G.Pisier, The Operator Hilbert spaceOH, Complex Interpolation and Tensor Norms. To appear, in Mem. Amer. Math. Soc.
[15] G.Pisier, Factorization of linear operators and the Geometry of Banach spaces CBMS (Regional conferences of the A.M.S.) 60, (1986), Reprinted with corrections 1987. · Zbl 0588.46010
[16] H. H.Schaefer, Banach lattices and positive operators. Berlin-Heidelberg-New York 1974. · Zbl 0296.47023
[17] L. Weiss, Integral operators and changes of density. Indiana Univ. Math. J.31, 83-96 (1982). · Zbl 0492.47017 · doi:10.1512/iumj.1982.31.31010
[18] Q. Xu, Notes on interpolation of Hardy spaces. Ann. Inst. Fourier (Grenoble)42, 875-889 (1992). · Zbl 0760.46060
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