On the convergence of iterates of convolution operators in Banach spaces
DOI:
https://doi.org/10.7146/math.scand.a-119601Abstract
Let $G$ be a locally compact abelian group and let $M(G)$ be the measure algebra of $G$. A measure $\mu \in M(G)$ is said to be power bounded if $\sup _{n\geq 0}\lVert \mu ^{n} \rVert _{1}<\infty $. Let $\mathbf {T} = \{ T_{g}:g\in G\}$ be a bounded and continuous representation of $G$ on a Banach space $X$. For any $\mu \in M(G)$, there is a bounded linear operator on $X$ associated with µ, denoted by $\mathbf {T}_{\mu }$, which integrates $T_{g}$ with respect to µ. In this paper, we study norm and almost everywhere behavior of the sequences $\{ \mathbf {T}_{\mu }^{n}x\}$ $(x\in X)$ in the case when µ is power bounded. Some related problems are also discussed.
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Arveson, W., The harmonic analysis of automorphism groups, in “Operator algebras and applications, Part I (Kingston, Ont., 1980)”, Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 199–269.
Bellow, A., Jones, R., and Rosenblatt, J., Almost everywhere convergence of powers, in “Almost everywhere convergence (Columbus, OH, 1988)”, Academic Press, Boston, MA, 1989, pp. 99–120.
Bellow, A., Jones, R., and Rosenblatt, J., Almost everywhere convergence of convolution powers, Ergodic Theory Dynam. Systems 14 (1994), no. 3, 415–432. https://doi.org/10.1017/S0143385700007951
Cohen, G., Cuny, C., and Lin, M., Almost everywhere convergence of powers of some positive $L_p$ contractions, J. Math. Anal. Appl. 420 (2014), no. 2, 1129–1153. https://doi.org/10.1016/j.jmaa.2014.06.014
Conze, J.-P. and Lin, M., Almost everywhere convergence of convolution powers on compact Abelian groups, Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), no. 2, 550–568. https://doi.org/10.1214/11-aihp468
Derriennic, Y. and Lin, M., Convergence of iterates of averages of certain operator representations and of convolution powers, J. Funct. Anal. 85 (1989), no. 1, 86–102. https://doi.org/10.1016/0022-1236(89)90047-5
Dunford, N. and Schwartz, J. T., Linear Operators. I. General Theory, Pure and Applied Mathematics, vol. 7, Interscience Publishers, New York, London, 1958.
Foguel, S. R., On iterates of convolutions, Proc. Amer. Math. Soc. 47 (1975), 368–370. https://doi.org/10.2307/2039748
Granirer, E. E., On some properties of the Banach algebras $A_p(G)$ for locally compact groups, Proc. Amer. Math. Soc. 95 (1985), no. 3, 375–381. https://doi.org/10.2307/2045806
Jones, R., Rosenblatt, J., and Tempelman, A., Ergodic theorems for convolutions of a measure on a group, Illinois J. Math. 38 (1994), no. 4, 521–553.
Kaniuth, E., Lau, A. T., and Ülger, A., Multipliers of commutative Banach algebras, power boundedness and Fourier-Stieltjes algebras, J. Lond. Math. Soc. (2) 81 (2010), no. 1, 255–275. https://doi.org/10.1112/jlms/jdp068
Krengel, U., Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. https://doi.org/10.1515/9783110844641
Larsen, R., Banach algebras: an introduction, Pure and Applied Mathematics, no. 24, Marcel Dekker, Inc., New York, 1973.
Laursen, K. B. and Neumann, M. M., An introduction to local spectral theory, London Mathematical Society Monographs. New Series, vol. 20, The Clarendon Press, Oxford, 2000.
Mustafayev, H., Distance formulas in group algebras, C. R. Math. Acad. Sci. Paris 354 (2016), no. 6, 577–582. https://doi.org/10.1016/j.crma.2016.04.002
Mustafayev, H., Convergence of iterates of convolution operators in $L^p$ spaces, Bull. Sci. Math. 152 (2019), 61–92. https://doi.org/10.1016/j.bulsci.2019.01.005
Nagy, B. and Zemánek, J., A resolvent condition implying power boundedness, Studia Math. 134 (1999), no. 2, 143–151. https://doi.org/10.4064/sm-134-2-143-151
Rudin, W., Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, no. 12, Interscience Publishers, New York-London, 1962.
Schreiber, B. M., Measures with bounded convolution powers, Trans. Amer. Math. Soc. 151 (1970), 405–431. https://doi.org/10.2307/1995504
Stam, A. J., On shifting iterated convolutions. I, Compositio Math. 17 (1966), 268–280.
Stein, E. M., On the maximal ergodic theorem, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1894–1897. https://doi.org/10.1073/pnas.47.12.1894
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Published
2020-05-06
How to Cite
Mustafayev, H. (2020). On the convergence of iterates of convolution operators in Banach spaces. MATHEMATICA SCANDINAVICA, 126(2), 339–366. https://doi.org/10.7146/math.scand.a-119601
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