Abstract
We introduce Duhamel algebras and study their properties and applications. We prove that a Banach space of analytic functions on the unit disc that satisfy certain conditions is a Duhamel algebra and describe its closed ideals. These results substantially generalize and improve the main results of Wigley’s papers. Some other related questions are also discussed.
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References
A. Aleman and B. Korenblum, “Volterra invariant subspaces of H p,” Bull. Sci. Math., 132:6 (2008), 510–528.
A. Biswas, A. Lambert, and S. Petrovic, “Extended eigenvalues and the Volterra operator,” Glasgow Math. J., 44:3 (2002), 521–534.
I. Yu. Domanov and M. M. Malamud, “On the spectral analysis of direct sums of Riemann–Liouville operators in Sobolev spaces of vector functions,” Integral Equations Operator Theory, 63:2 (2009), 181–215.
H. Guediri, M. T. Garayev, and H. Sadraoui, “The Bergman space as a Banach algebra,” New York J. Math., 21 (2015), 339–350.
B. Hollenbeck and I. E. Verbitsky, “Best constants for the Riesz projection,” J. Funct. Anal., 175:2 (2000), 370–392.
M. T. Karaev, “Invariant subspaces, cyclic vectors, commutant and extended eigenvectors of some convolution operators,” Methods Funct. Anal. Topology, 11:1 (2005), 48–59.
M. T. Karaev, M. Gürdal, and S. Saltan, “Some applications of Banach algebras techniques,” Math. Nachr., 284:13 (2011), 1678–1689.
M. T. Karaev, “On extended eigenvalues and extended eigenvectors of some operator classes,” Proc. Amer. Math. Soc., 134:6 (2006), 2383–2392.
Yu. Linchuk, “On derivation operators with respect to the Duhamel convolution in the space of analytic functions,” Math. Commun., 20:1 (2015), 17–22.
M. M. Malamud, “Invariant and hyperinvariant subspaces of direct sums of simple Volterra operators,” in: Oper. Theory Adv. Appl., vol. 102, BirkhaÜser, Basel, 1998, 143–167.
M. M. Malamud, “Similarity of Volterra operators and related questions of the theory of differential equations of fractional orders,” Trudy Moscov. Mat. Obsch., 55 (1994), 73–148; English transl.: Trans. Moscow Math. Soc., 55 (1994), 57–122.
K. G. Merryfield and S. Watson, “A local algebra structure for H p of the polydisc,” Colloq. Math., 62:1 (1991), 73–79.
N. K. Nikolski, “Selected problems of weighted approximation and spectral analysis [in Russian],” Trudy Mat. Inst. Steklov., 70 (1974), 1–270.
N. M. Wigley, “The Duhamel product of analytic functions,” Duke Math. J., 41 (1974), 211–217.
N. M. Wigley, “A Banach algebra structure for H p,” Canad. Math. Bull., 18 (1975), 597–603.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 52, No. 1, pp. 3–12, 2018
Original Russian Text Copyright © by M. T. Karaev
This work was financially supported by the Deanship of Scientific Research, King Saud University, in the framework of the Research Group Project no. RGP-VPP-323.
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Karaev, M.T. Duhamel Algebras and Applications. Funct Anal Its Appl 52, 1–8 (2018). https://doi.org/10.1007/s10688-018-0201-z
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DOI: https://doi.org/10.1007/s10688-018-0201-z