Abstract
We consider generalized Hausdorff operators and introduce the notion of the symbol of such an operator. Using this notion, we describe, under some natural conditions, the structure and investigate important properties (such as invertibility, spectrum, and norm) of normal generalized Hausdorff operators on Lebesgue spaces over ℝn. As an example we consider Cesàro operators.
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References
A. Brown, P. R. Halmos, and A. L. Shields, “Cesaro operators,” Acta Sci. Math. (Szeged), 26 (1965), 125–137.
G. Brown and F. Moricz, “Multivariate Hausdorff operators on the spaces Lp(ℝn),” J. Math. Anal. Appl., 271:2 (2002), 443–454.
Yu. A. Brychkov, H.-J. Glaeske, A. P. Prudnikov, and Vu Kim Tuan, Multidimensional Integral Transformations, Gordon and Breach Science Publishers, Philadelphia-Reading-Paris-Montreux-Tokyo-Melbourne, 1992.
Jie-cheng Chen, Da-shan Fan, and Si-lei Wang, “Hausdorff operators on Euclidean space (a survey article),” Appl. Math. J. Chinese Univ. Ser. B, 28:4 (2013), 548–564.
C. Georgakis, “The Hausdorff mean of a Fourier-Stieltjes transform,” Proc. Amer. Math. Soc., 116:2 (1992), 465–471.
G. H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949.
C. S. Kubrusly, Elements of Operator Theory, Birkhauser, Boston, 2001.
A. Lerner and E. Liflyand, “Multidimensional Hausdorff operators on the real Hardy space,” J. Austr. Math. Soc., 83:1 (2007), 79–86.
E. Liflyand, “Open problems on Hausdorff operators,” in: Complex Analysis and Potential Theory, Proc. Conf. Satellite to ICM 2006 (Gebze, Turkey, 8–14 Sept. 2006), World Scientific, Hackensack, 2007, 280–285.
E. Liflyand, “Hausdorff operators on Hardy spaces,” Eurasian Math. J., 4:4 (2013), 101–141.
E. Liflyand and F. Móricz, “The Hausdorff operator is bounded on the real Hardy space H1(ℝ),” Proc. Amer. Math. Soc., 128:5 (2000), 1391–1396.
A. R. Mirotin, “Boundedness of Hausdorff operators on real Hardy spaces H1 over locally compact groups,” J. Math. Anal. Appl., 473 (2019), 519–533; https://arxiv.org/abs/1808.08257v2.
A. R. Mirotin, “Hilbert transform in context of locally compact abelian groups,” Int. J. Pure Appl. Math., 51:4 (2009), 463–474.
A. R. Mirotin, The structure of normal Hausdorff operators on Lebesgue spaces, https://arxiv.org/abs/1812.02680v2.
B. E. Rhoades, “Spectra of some Hausdorff operators,” Acta. Sci. Math. (Szeged), 32 (1971), 91–100.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. Part II. The Transcendental Functions, Cambridge University Press, Cambridge, 1927.
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The author thanks the referee for quickly and carefully reading the manuscript and comments that have helped to improve the presentation.
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Dedicated to the memory of my teacher Evgeniĭ Alekseevich Gorin
Russian Text © The Author(s), 2019, published in Funktsional’nyi Analiz i Ego Prilozheniya, 2019, Vol. 53, No. 4, pp.27–37.
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Mirotin, A.R. On the Structure of Normal Hausdorff Operators on Lebesgue Spaces. Funct Anal Its Appl 53, 261–269 (2019). https://doi.org/10.1134/S0016266319040038
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DOI: https://doi.org/10.1134/S0016266319040038