Abstract
Let E be a symmetric Banach space with the Fatou property and \(1<p_E\le q_E<p\). We prove the duality for symmetric Banach space \(_p\widehat{E}(\mathcal {M})\) which is a kind of noncommutative quasi-martingale space. As its applications, we discuss concrete description of the symmetric Banach space \(_p\widehat{E}(\mathcal {M})\) as interpolations of quasi-martingale \(L_p\)-spaces.
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References
T. N. Bekjan, Duality for symmetric Hardy spaces of noncommutative martingales, Math. Zeit. 289 (2018), 787-802.
C. Bennett, Banach function spaces and interpolation methods. I. The abstract theory, J. Funct. Anal. 17 (1974), 409-440.
C. Bennett, Banach function spaces and interpolation methods. II. Interpolation of weak-type operators, Linear operators and approximation, II (Proc. Conf., Math. Res. Inst., Oberwolfach, 1974), Birkh auser, Basel, 1974, pp. 129-139. Internat. Ser. Numer. Math., Vol. 25.
I. Cuculescu, Martingales on von neumann algebra, J. Multivariate Anal. 1 (1971), 17- 27.
P. G. Dodds, T. K. Dodds and B. de Pagter, Noncommutative Banach function spaces, Math. Z. 201 (1989), 583-579.
P. G. Dodds, T. K. Dodds and B. de Pagter, Noncommutative K\({\ddot{\text{ o }}}\)the duality, Trans. Amer. Math. Soc. 339 (1993), 717-750.
P. G. Dodds, T. K. Dodds and B. de Pager, Fully symmetric operator spaces, Integr. Equ. Oper. Theory. 15 (1992), 942-972.
S. Dirksen, Noncommutative Boyd interpolation theorems, Trans. Amer. Math. Soc. 367 (2015), 4079-4110.
Y. Jiao, Martingale inequalities in noncommutative symmetric spaces, Arch. Math. (Basel), 98 (2012) 87-97.
M. Junge, Doobs inequality for non-commutative martingales, J. Reine Angew. Math, 549 (2002), 149-190.
Pawel Kolwicz, Karol Le snik and Lech Maligranda, Pointwise products of some Banach function spaces and factorization, J. Funct. Anal, 266 (2014), 616-659.
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II, Springer-Verlag, Berlin, 1979.
P. Liu, UMD spaces and the laws of large numbers of Banach spaces-valued martingales, Chin. Bull. Sci. 34 (1989), 401-404.
F.Lust-Piquard and G.Pisier, Non-commutative Khintchine and Paley inequalities, Ark. Mat. 29 (1991), 241-260.
C. Ma and Y. Hou, Duality theorems for noncommutative quasimartingale spaces, Acta Math. Hungar. 148 (2016), 132-146.
C. Le Merdy and F. Sukochev, Radermacher averages on noncommutative symmetric spaces, J. Funct. Anal. 255 (2008), 3329-3355.
G. Pisier and Q. Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997), 667-698.
G. Pisier and Q. Xu, Non-Commutative \(L_p\)-Spaces, Handbook of the geometry of Banach spaces, Vol. 2, 1459-1517, North-Holland, Amsterdam, (2003).
N. Randrianantoanina and L. Wu, Martingale inequalities in noncommutative symmetric spaces, J. Funct. Anal. 269 (2015), 2222-2253.
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This work was supported by National Natural Science Foundation of China (11801489,11671308).
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Communicated by Gadadhar Misra.
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Ma, C., Fan, L., Zhang, X. et al. Duality and interpolation for symmetric Banach spaces of noncommutative quasi-martingales. Indian J Pure Appl Math 54, 630–640 (2023). https://doi.org/10.1007/s13226-022-00281-2
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DOI: https://doi.org/10.1007/s13226-022-00281-2