Previous |  Up |  Next


holomorphic function; Banach algebra; generator
Pseudoconvex domains are exhausted in such a way that we keep a part of the boundary fixed in all the domains of the exhaustion. This is used to solve a problem concerning whether the generators for the ideal of either the holomorphic functions continuous up to the boundary or the bounded holomorphic functions, vanishing at a point in $\mathbb{C}^{n}$ where the fibre is nontrivial, has to exceed $n$. This is shown not to be the case.
[1] Kenzō Adachi: Continuation of bounded holomorphic functions from certain subvarieties to weakly pseudoconvex domains. Pacific J. Math. 130 (1987), 1–8. MR 0910650
[2] Ulf Backlund, Anders Fällström: Counterexamples to the Gleason problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1998), 595–603. MR 1635710
[3] Linus Carlsson, Urban Cegrell, Anders Fällström: Spectrum of certain Banach algebras and $\overline{\partial }$ problems. Ann. Pol. Math. 90 (2007), 51–58. MR 2283112
[4] So-Chin Chen, Mei-Chi Shaw: Partial Differential Equations in Several Complex Variables, vol. 19 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI, 2001. MR 1800297
[5] Klas Diederich, John Erik Fornaess: Pseudoconvex domains: an example with nontrivial nebenhülle. Math. Ann. 225 (1977), 275–292. MR 0430315
[6] Andrew M. Gleason: Finitely generated ideals in Banach algebras. J. Math. Mech. 13 (1964), 125–132. MR 0159241
[7] Pengfei Guan: The extremal function associated to intrinsic norms. Ann. Math. 156 (2002), 197–211. MR 1935845
[8] Gennadi Henkin, Jürgen Leiterer: Theory of Functions on Complex Manifolds. Monographs in Mathematics vol. 79, Birkhäuser, Basel, 1984. MR 0774049
[9] Marek Jarnicki, Peter Pflug: Extension of Holomorphic Functions. Expositions in Mathematics vol. 34, Walter de Gruyter, Berlin, 2000. MR 1797263
[10] Nils Øvrelid: Generators of the maximal ideals of $A(\bar{D})$. Pacific J. Math. 39 (1971), 219–223. MR 0310292
Partner of
EuDML logo