## Harmonic functions of maximal growth: invertibility and cyclicity in Bergman spaces

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- by Alexander Borichev and Hå kan Hedenmalm
- J. Amer. Math. Soc.
**10**(1997), 761-796 - DOI: https://doi.org/10.1090/S0894-0347-97-00244-0
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## Abstract:

In the theory of commutative Banach algebras with unit, an element generates a dense ideal if and only if it is invertible, in which case its Gelfand transform has no zeros, and the ideal it generates is the whole algebra. With varying degrees of success, efforts have been made to extend the validity of this result beyond the context of Banach algebras. For instance, for the Hardy space $H^{2}$ on the unit disk, it is known that all invertible elements are cyclic (an element is cyclic if its polynomial multiples are dense), but cyclic elements need not be invertible. In this paper, we supply examples of functions in the Bergman and uniform Bergman spaces on the unit disk which are invertible, but not cyclic. This answers in the negative questions raised by Shapiro, Nikolskiĭ, Shields, Korenblum, Brown, and Frankfurt.## References

- D. Aharonov, H. S. Shapiro, and A. L. Shields,
*Weakly invertible elements in the space of square-summable holomorphic functions*, J. London Math. Soc. (2)**9**(1974/75), 183–192. MR**365150**, DOI 10.1112/jlms/s2-9.1.183 - A. Aleman, S. Richter, C. Sundberg,
*Beurling’s theorem for the Bergman space*, preprint 1995. - A. A. Borichev and P. J. H. Hedenmalm,
*Cyclicity in Bergman-type spaces*, Internat. Math. Res. Notices**5**(1995), 253–262. MR**1333752**, DOI 10.1155/S1073792895000201 - Leon Brown and Boris Korenblum,
*Cyclic vectors in $A^{-\infty }$*, Proc. Amer. Math. Soc.**102**(1988), no. 1, 137–138. MR**915731**, DOI 10.1090/S0002-9939-1988-0915731-9 - Leon Brown, Allen Shields, and Karl Zeller,
*On absolutely convergent exponential sums*, Trans. Amer. Math. Soc.**96**(1960), 162–183. MR**142763**, DOI 10.1090/S0002-9947-1960-0142763-8 - Nelson Dunford and Jacob T. Schwartz,
*Linear Operators. I. General Theory*, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR**0117523** - Peter Duren, Dmitry Khavinson, Harold S. Shapiro, and Carl Sundberg,
*Contractive zero-divisors in Bergman spaces*, Pacific J. Math.**157**(1993), no. 1, 37–56. MR**1197044**, DOI 10.2140/pjm.1993.157.37 - P. Duren, D. Khavinson, H. S. Shapiro, and C. Sundberg,
*Invariant subspaces in Bergman spaces and the biharmonic equation*, Michigan Math. J.**41**(1994), no. 2, 247–259. MR**1278431**, DOI 10.1307/mmj/1029004992 - V. P. Havin and N. K. Nikolski (eds.),
*Linear and complex analysis. Problem book 3. Part I*, Lecture Notes in Mathematics, vol. 1573, Springer-Verlag, Berlin, 1994. MR**1334345** - 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** - Håkan Hedenmalm,
*A factorization theorem for square area-integrable analytic functions*, J. Reine Angew. Math.**422**(1991), 45–68. MR**1133317**, DOI 10.1515/crll.1991.422.45 - Per Jan Håkan Hedenmalm,
*Open problems in the function theory of the Bergman space*, Festschrift in honour of Lennart Carleson and Yngve Domar (Uppsala, 1993) Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., vol. 58, Uppsala Univ., Uppsala, 1995, pp. 153–169. MR**1352006** - Boris Korenblum,
*An extension of the Nevanlinna theory*, Acta Math.**135**(1975), no. 3-4, 187–219. MR**425124**, DOI 10.1007/BF02392019 - Boris Korenblum,
*A Beurling-type theorem*, Acta Math.**138**(1976), no. 3-4, 265–293. MR**447584**, DOI 10.1007/BF02392318 - V. P. Havin and N. K. Nikolski (eds.),
*Linear and complex analysis. Problem book 3. Part I*, Lecture Notes in Mathematics, vol. 1573, Springer-Verlag, Berlin, 1994. MR**1334345** - Boris Korenblum,
*Outer functions and cyclic elements in Bergman spaces*, J. Funct. Anal.**115**(1993), no. 1, 104–118. MR**1228143**, DOI 10.1006/jfan.1993.1082 - 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** - Kristian Seip,
*Beurling type density theorems in the unit disk*, Invent. Math.**113**(1993), no. 1, 21–39. MR**1223222**, DOI 10.1007/BF01244300 - F. A. Shamoyan,
*Weak invertibility in some spaces of analytic functions*, Akad. Nauk Armyan. SSR Dokl.**74**(1982), no. 4, 157–161 (Russian, with Armenian summary). MR**671961** - Harold S. Shapiro,
*Weighted polynomial approximation and boundary behavior of analytic functions*, Contemporary Problems in Theory Anal. Functions (Internat. Conf., Erevan, 1965) Izdat. “Nauka”, Moscow, 1966, pp. 326–335. MR**0209485** - G. Šapiro,
*Some observations concerning weighted polynomial approximation of holomorphic functions*, Mat. Sb. (N.S.)**73 (115)**(1967), 320–330 (Russian). MR**0217304** - Allen L. Shields,
*Weighted shift operators and analytic function theory*, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR**0361899** - V. P. Havin and N. K. Nikolski (eds.),
*Linear and complex analysis. Problem book 3. Part I*, Lecture Notes in Mathematics, vol. 1573, Springer-Verlag, Berlin, 1994. MR**1334345** - Allen L. Shields,
*Cyclic vectors in Banach spaces of analytic functions*, Operators and function theory (Lancaster, 1984) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 153, Reidel, Dordrecht, 1985, pp. 315–349. MR**810450**

## Bibliographic Information

**Alexander Borichev**- Affiliation: Department of Mathematics, University of Bordeaux I, 351, cours de la Liberation, 33405 Talence, France
- Email: borichev@math.u-bordeaux.fr
**Hå kan Hedenmalm**- Affiliation: Department of Mathematics, Lund University, Box 118, 22100 Lund, Sweden
- Email: haakan@maths.lth.se
- Received by editor(s): July 18, 1996
- Received by editor(s) in revised form: March 25, 1997
- Additional Notes: The research of both authors was supported in part by the Swedish Natural Science Research Council. The second author was also supported by the 1992 Wallenberg Prize from the Swedish Mathematical Society.

The paper was written when the authors worked at the Uppsala University, Sweden, and visited MSRI, USA - © Copyright 1997 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**10**(1997), 761-796 - MSC (1991): Primary 30H05, 46E15
- DOI: https://doi.org/10.1090/S0894-0347-97-00244-0
- MathSciNet review: 1446365