Uniform Convergence on Subspaces in von Neumann Ergodic
Matematičeskie zametki, Tome 113 (2023) no. 5, pp. 713-730.

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We consider the power-law uniform (in the operator norm) convergence on vector subspaces with their own norms in the von Neumann ergodic theorem with discrete time. All possible exponents of the considered power-law convergence are found; for each of these exponents, spectral criteria for such convergence are given and the complete description of all such subspaces is obtained. Uniform convergence on the whole space takes place only in the trivial cases, which explains the interest in uniform convergence precisely on subspaces. In addition, by the way, old estimates of the rates of convergence in the von Neumann ergodic theorem for measure-preserving mappings are generalized and refined.
Keywords: von Neumann ergodic theorem , rate of convergence in ergodic theorems, power-law uniform convergence.
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A. G. Kachurovskii; I. V. Podvigin; A. J. Khakimbaev. Uniform Convergence on Subspaces in von Neumann Ergodic. Matematičeskie zametki, Tome 113 (2023) no. 5, pp. 713-730. http://geodesic.mathdoc.fr/item/MZM_2023_113_5_a6/

[1] N. I. Akhiezer, I. M. Glazman, Teoriya lineinykh operatorov v gilbertovom prostranstve, Nauka, M., 1966 | MR

[2] N. K. Bari, Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR

[3] I. A. Ibragimov, Yu. V. Linnik, Nezavisimye i statsionarno svyazannye velichiny, Nauka, M., 1965 | MR

[4] J. Ben-Artzi, B. Morisse, “Uniform convergence in von Neumann's ergodic theorem in the absence of a spectral gap”, Ergodic Theory Dynam. Systems, 41:6 (2021), 1601–1611 | DOI | MR

[5] A. G. Kachurovskii, I. V. Podvigin, V. E. Todikov, “Uniform convergence on subspaces in von Neumann's ergodic theorem with continuous time”, Siberian Electronic Math. Reports, 20:1 (2023), 183–206

[6] A. G. Kachurovskii, I. V. Podvigin, “Otsenki skorostei skhodimosti v ergodicheskikh teoremakh fon Neimana i Birkgofa”, Tr. MMO, 77:1 (2016), 1–66 | MR

[7] A. A. Prikhodko, “Ergodicheskie avtomorfizmy s prostym spektrom i svoistvom bystrogo ubyvaniya korrelyatsii”, Matem. zametki, 94:6 (2013), 949–954 | DOI | MR | Zbl

[8] V. V. Ryzhikov, “Ergodicheskie gomoklinicheskie gruppy, sidonovskie konstruktsii i puassonovskie nadstroiki”, Tr. MMO, 75:1 (2014), 93–103

[9] A. A. Prikhodko, “Ob ergodicheskikh potokakh s prostym lebegovskim spektrom”, Matem. sb., 211:4 (2020), 123–144 | DOI

[10] A. G. Kachurovskii, V. V. Sedalischev, “O konstantakh otsenok skorosti skhodimosti v ergodicheskoi teoreme fon Neimana”, Matem. zametki, 87:5 (2010), 756–763 | DOI | MR

[11] A. G. Kachurovskii, V. V. Sedalischev, “Konstanty otsenok skorosti skhodimosti v ergodicheskikh teoremakh fon Neimana i Birkgofa”, Matem. sb., 202:8 (2011), 21–40 | DOI | MR | Zbl

[12] A. G. Kachurovskii, “Skorosti skhodimosti v ergodicheskikh teoremakh”, UMN, 51:4 (310) (1996), 73–124 | DOI | MR | Zbl

[13] V. F. Gaposhkin, “O skorosti ubyvaniya veroyatnostei $\varepsilon$-uklonenii srednikh statsionarnykh protsessov”, Matem. zametki, 64:3 (1998), 366–372 | DOI | MR | Zbl

[14] U. Rudin, Osnovy matematicheskogo analiza, Mir, M., 1976 | MR

[15] A. N. Shiryaev, Veroyatnost, Nauka, M., 1989 | MR

[16] F. A. Robinson, “Sums of stationary random variables”, Proc. Amer. Math. Soc., 11:1 (1960), 77–79 | DOI | MR

[17] P. L. Butzer, U. Westphal, “The mean ergodic theorem and saturation”, Indiana Univ. Math. J., 20 (1971), 1163–1174 | DOI | MR

[18] V. F. Gaposhkin, “Skhodimost ryadov, svyazannykh so statsionarnymi posledovatelnostyami”, Izv. AN SSSR. Ser. matem., 39:6 (1975), 1366–1392 | MR | Zbl

[19] V. V. Sedalischev, “Svyaz skorostei skhodimosti v ergodicheskikh teoremakh fon Neimana i Birkgofa v $L_p$”, Sib. matem. zhurn., 55:2 (2014), 412–426 | MR

[20] F. E. Browder, “On the iteration of transformations in noncompact minimal dynamical systems”, Proc. Amer. Math. Soc., 9 (1958), 773–780 | DOI | MR

[21] M. Lin, “On the uniform ergodic theorem”, Proc. Amer. Math. Soc., 43:2 (1974), 337–340 | DOI | MR

[22] E. Yu. Emel'yanov, Non-Spectral Asymptotic Analysis of One-Parameter Operator Semigroups, Operator Theory: Adv. and Appl., 173, Birkhäuser, Basel, 2007 | MR

[23] A. Gomilko, M. Haase, Yu. Tomilov, “On rates in mean ergodic theorems”, Math. Res. Lett., 18:2 (2011), 201–213 | DOI | MR

[24] T. Eisner, “Embedding operators into strongly continuous semigroups”, Arch. Math., 92:5 (2009), 451–460 | DOI | MR

[25] A. M. Stepin, A. M. Eremenko, “Needinstvennost vklyucheniya v potok i obshirnost tsentralizatora dlya tipichnogo sokhranyayuschego meru preobrazovaniya”, Matem. sb., 195:12 (2004), 95–108 | DOI | MR | Zbl

[26] I. P. Kornfeld, Ya. G. Sinai, S. V. Fomin, Ergodicheskaya teoriya, Nauka, M., 1980 | MR