Cyclicity of nonvanishing functions in the polydisk and in the ball

Amar, E.; Thomas, P.

doi:10.1090/spmj/1622

Cyclicity of nonvanishing functions in the polydisk and in the ball
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by E. Amar and P. J. Thomas

St. Petersburg Math. J. 31 (2020), 751-768

DOI: https://doi.org/10.1090/spmj/1622

Published electronically: September 3, 2020

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Abstract:

A special version of the corona theorem in several variables, valid when all but one of the data functions are smooth, is used to generalize, to the polydisk and to the ball, the results obtained by El Fallah, Kellay, and Seip about the cyclicity of nonvanishing bounded holomorphic functions in sufficiently large Banach spaces of analytic functions determined either by weighted sums of powers of Taylor coefficients or by radially weighted integrals of powers of the modulus of the function.

References

Éric Amar, A problem of ideals, Astérisque 217 (1993), 5, 9–12. Colloque d’Analyse Complexe et Géométrie (Marseille, 1992). MR 1247747
E. Amar and C. Menini, Universal divisors in Hardy spaces, Studia Math. 143 (2000), no. 1, 1–21. MR 1814477, DOI 10.4064/sm-143-1-1-21
Alexander Borichev and Håkan Hedenmalm, Completeness of translates in weighted spaces on the half-line, Acta Math. 174 (1995), no. 1, 1–84. MR 1310845, DOI 10.1007/BF02392801
Urban Cegrell, Representing measures in the spectrum of $H^\infty (\Omega )$, Complex analysis (Wuppertal, 1991) Aspects Math., E17, Friedr. Vieweg, Braunschweig, 1991, pp. 77–80. MR 1122163
Omar El-Fallah, Karim Kellay, and Kristian Seip, Cyclicity of singular inner functions from the corona theorem, J. Inst. Math. Jussieu 11 (2012), no. 4, 815–824. MR 2979822, DOI 10.1017/S1474748012000035
Michael Hartz, Von Neumann’s inequality for commuting weighted shifts, Indiana Univ. Math. J. 66 (2017), no. 4, 1065–1079. MR 3689326, DOI 10.1512/iumj.2017.66.6074
Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR 1758653, DOI 10.1007/978-1-4612-0497-8
Lars Hörmander, Generators for some rings of analytic functions, Bull. Amer. Math. Soc. 73 (1967), 943–949. MR 226387, DOI 10.1090/S0002-9904-1967-11860-3
Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
Steven G. Krantz, Function theory of several complex variables, 2nd ed., The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. MR 1162310
N. K. Nikol′skiĭ, Izbrannye zadachi vesovoĭ approksimatsii i spektral′nogo analiza, Izdat. “Nauka” Leningrad. Otdel., Leningrad, 1974 (Russian). Trudy Mat. Inst. Steklov. 120 (1974). MR 0467269
Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594