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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Outer functions in function algebras on the bidisc
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by Håkan Hedenmalm PDF
Trans. Amer. Math. Soc. 306 (1988), 697-714 Request permission

Abstract:

Let $f$ be a function in the bidisc algebra $A({{\mathbf {D}}^2})$ whose zero set $Z(f)$ is contained in $\{ 1\} \times \overline {\mathbf {D}}$. We show that the closure of the ideal generated by $f$ coincides with the ideal of functions vanishing on $Z(f)$ if and only if $f( \cdot , \alpha )$ is an outer function for all $\alpha \in \overline {\mathbf {D}}$, and $f(1, \cdot )$ either vanishes identically or is an outer function. Similar results are obtained for a few other function algebras on ${{\mathbf {D}}^2}$ as well.
References
  • O. P. Agrawal, D. N. Clark, and R. G. Douglas, Invariant subspaces in the polydisk, Pacific J. Math. 121 (1986), no. 1, 1–11. MR 815027
  • Colin Bennett and John E. Gilbert, Homogeneous algebras on the circle. I. Ideals of analytic functions, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 3, 1–19 (English, with French summary). MR 338782
    • T. Carleman, Les fonctions quasi-analytiques, Gauthier-Villars, Paris, 1926.
  • Yngve Domar, On the analytic transform of bounded linear functionals on certain Banach algebras, Studia Math. 53 (1975), no. 3, 203–224. MR 383003, DOI 10.4064/sm-53-3-203-224
    • T. W. Gamelin, Lectures on ${H^\infty }(D)$, Notas de Mathematica, No. 21, Universidad Nacional de La Plata, La Plata, Argentina, 1972.
  • John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
    • I. M. Gelfand, D. A. Raǐkov and G. E. Šilov, Commutative normed rings, Chelsea, Bronx, N.Y., 1964.
  • V. P. Gurariĭ, Harmonic analysis in spaces with weight, Trudy Moskov. Mat. Obšč. 35 (1976), 21–76 (Russian). MR 0499942
  • Håkan Hedenmalm, On the primary ideal structure at infinity for analytic Beurling algebras, Ark. Mat. 23 (1985), no. 1, 129–158. MR 800176, DOI 10.1007/BF02384421
  • Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
    • M. R. Janeba, Analytic structures in certain compactifications of the unit ball and polydisc in ${{\mathbf {C}}^2}$, Dissertation, Univ. of California, Santa Barbara, 1983. P. Koosis, Introduction to ${H_p}$ spaces, London Math. Soc. Lecture Notes 40 (1980).
  • B. I. Korenbljum, Invariant subspaces of the shift operator in a weighted Hilbert space, Mat. Sb. (N.S.) 89(131) (1972), 110–137, 166 (Russian). MR 0317073
    • —, Closed ideals in the ring ${A^n}$, Functional Anal. Appl. 6 (1972), 203-214. B. Ya. Levin, Translates of functions of two variables, Problem 7.20, Linear and Complex Analysis Problem Book, 199 Research Problems, Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1984, p. 421.
  • Bertil Nyman, On the One-Dimensional Translation Group and Semi-Group in Certain Function Spaces, University of Uppsala, Uppsala, 1950. Thesis. MR 0036444
  • Jean-Pierre Rosay, Une équivalence au corona problem dans C$^{n}$ et un problème d’idéal dans $H^{\infty }\,(D)$, J. Functional Analysis 7 (1971), 71–84 (French). MR 0273413, DOI 10.1016/0022-1236(71)90045-0
  • L. A. Rubel and A. L. Shields, The failure of interior-exterior factorization in the polydisc and the ball, Tohoku Math. J. (2) 24 (1972), 409–413. MR 333234, DOI 10.2748/tmj/1178241479
  • Walter Rudin, The closed ideals in an algebra of analytic functions, Canadian J. Math. 9 (1957), 426–434. MR 89254, DOI 10.4153/CJM-1957-050-0
  • Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
  • Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043
  • B. A. Taylor and D. L. Williams, Ideals in rings of analytic functions with smooth boundary values, Canadian J. Math. 22 (1970), 1266–1283. MR 273024, DOI 10.4153/CJM-1970-143-x
  • Philippe Tchamitchian, Généralisation des algèbres de Beurling, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 4, 151–168 (French, with English summary). MR 766278
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 306 (1988), 697-714
  • MSC: Primary 32A35; Secondary 32E25, 46J15
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0933313-4
  • MathSciNet review: 933313