Corona-Theorem

Corona theorem In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by Kakutani (1941) and proved by Lennart Carleson (1962).

The commutative Banach algebra and Hardy space H∞ consists of the bounded holomorphic functions on the open unit disc D. Its spectrum S (the closed maximal ideals) contains D as an open subspace because for each z in D there is a maximal ideal consisting of functions f with f(z) = 0.

The subspace D cannot make up the entire spectrum S, essentially because the spectrum is a compact space and D is not. The complement of the closure of D in S was called the corona by Newman (1959), and the corona theorem states that the corona is empty, or in other words the open unit disc D is dense in the spectrum. A more elementary formulation is that elements f1,...,fn generate the unit ideal of H∞ if and only if there is some δ>0 such that {Anzeigestil |f_{1}|+cdots +|f_{n}|geq delta } everywhere in the unit ball.

Newman showed that the corona theorem can be reduced to an interpolation problem, which was then proved by Carleson.

Im 1979 Thomas Wolff gave a simplified (but unpublished) proof of the corona theorem, described in (Koosis 1980) und (Gamelin 1980).

Cole later showed that this result cannot be extended to all open Riemann surfaces (Gamelin 1978).

As a by-product, of Carleson's work, the Carleson measure was invented which itself is a very useful tool in modern function theory. It remains an open question whether there are versions of the corona theorem for every planar domain or for higher-dimensional domains.

Note that if one assumes the continuity up to the boundary in Corona's theorem, then the conclusion follows easily from the theory of commutative Banach algebra (Rudin 1991).

See also Corona set References Carleson, Lennart (1962), "Interpolations by bounded analytic functions and the corona problem", Annalen der Mathematik, 76 (3): 547–559, doi:10.2307/1970375, JSTOR 1970375, HERR 0141789, Zbl 0112.29702 Gamelin, T. W. (1978), Uniform algebras and Jensen measures., Vorlesungsreihe der London Mathematical Society, vol. 32, Cambridge-New York: Cambridge University Press, pp. iii+162, ISBN 978-0-521-22280-8, HERR 0521440, Zbl 0418.46042 Gamelin, T. W. (1980), "Wolff's proof of the corona theorem", Israelisches Journal für Mathematik, 37 (1–2): 113–119, doi:10.1007/BF02762872, HERR 0599306, Zbl 0466.46050 Kakutani, Shizuo (1941). "Concrete representation of abstract (M)-spaces. (A characterization of the space of continuous functions.)". Ann. von Math. Serie 2. 42 (4): 994–1024. doi:10.2307/1968778. hdl:10338.dmlcz/100940. JSTOR 1968778. HERR 0005778. Koosis, Paul (1980), Introduction to Hp-spaces. With an appendix on Wolff's proof of the corona theorem, Vorlesungsreihe der London Mathematical Society, vol. 40, Cambridge-New York: Cambridge University Press, pp. xv+376, ISBN 0-521-23159-0, HERR 0565451, Zbl 0435.30001 Neuer Mann, D. J. (1959), "Some remarks on the maximal ideal structure of H∞", Annalen der Mathematik, 70 (2): 438–445, doi:10.2307/1970324, JSTOR 1970324, HERR 0106290, Zbl 0092.11802 Rudin, Walter (1991), Funktionsanalyse, p. 279. Schark, ich. J. (1961), "Maximal ideals in an algebra of bounded analytic functions", Journal of Mathematics and Mechanics, 10: 735–746, HERR 0125442, Zbl 0139.30402. verbergen vte Funktionsanalyse (Themen – Glossar) Leerzeichen BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraisch/topologisch)lokal konvexreflexivseparable Theoreme Hahn-BanachRiesz-Darstellunggeschlossener Graphgleichmäßiges BeschränktheitsprinzipKakutani-FixpunktKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operatoren adjointboundcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebren Banach-AlgebraC*-AlgebraSpektrum einer C*-AlgebraOperator-Algebravon Gruppenalgebra einer lokalvariant-kompakten Gruppe SubraumproblemMahlersche Vermutung Anwendungen Hardy-RaumSpektraltheorie gewöhnlicher DifferentialgleichungenWärmekernindexsatzVariationsrechnungFunktionsrechnungIntegraloperatorJones-PolynomTopologische QuantenfeldtheorieNichtkommutative GeometrieRiemann-HypotheseVerteilung (oder verallgemeinerte Funktionen) Fortgeschrittene Themen Approximation PropertyBalanced SetChoquet-TheorieSchwache TopologieBanach-Mazur-AbstandTomita-Takesaki-Theorie Kategorien: Banach algebrasHardy spacesTheorems in complex analysis