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 {style d'affichage |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.

Dans 1979 Thomas Wolff gave a simplified (but unpublished) proof of the corona theorem, described in (Koosis 1980) et (Gamelin 1980).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 (Roudine 1991).

See also Corona set References Carleson, Lennart (1962), "Interpolations by bounded analytic functions and the corona problem", Annales de Mathématiques, 76 (3): 547–559, est ce que je:10.2307/1970375, JSTOR 1970375, M 0141789, Zbl 0112.29702 Gamelin, J. O. (1978), Uniform algebras and Jensen measures., Série de notes de cours de la London Mathematical Society, volume. 32, Cambridge-New York: la presse de l'Universite de Cambridge, pp. iii+162, ISBN 978-0-521-22280-8, M 0521440, Zbl 0418.46042 Gamelin, J. O. (1980), "Wolff's proof of the corona theorem", Journal israélien de mathématiques, 37 (1–2): 113–119, est ce que je:10.1007/BF02762872, M 0599306, Zbl 0466.46050 Kakutani, Shizuo (1941). "Concrete representation of abstract (M)-spaces. (A characterization of the space of continuous functions.)". Anne. des mathématiques. Série 2. 42 (4): 994–1024. est ce que je:10.2307/1968778. hdl:10338.dmlcz/100940. JSTOR 1968778. M 0005778. Koosis, Paul (1980), Introduction to Hp-spaces. With an appendix on Wolff's proof of the corona theorem, Série de notes de cours de la London Mathematical Society, volume. 40, Cambridge-New York: la presse de l'Universite de Cambridge, pp. xv+376, ISBN 0-521-23159-0, M 0565451, Zbl 0435.30001 Homme nouveau, ré. J. (1959), "Some remarks on the maximal ideal structure of H∞", Annales de Mathématiques, 70 (2): 438–445, est ce que je:10.2307/1970324, JSTOR 1970324, M 0106290, Zbl 0092.11802 Roudine, Walter (1991), Analyse fonctionnelle, p. 279. Schark, je. J. (1961), "Maximal ideals in an algebra of bounded analytic functions", Journal de mathématiques et de mécanique, 10: 735–746, M 0125442, Zbl 0139.30402. cacher vte Analyse fonctionnelle (sujets – glossaire) Espaces BanachBesovFréchetHilbertHölderNucléaireOrliczSchwartzSobolevvecteur topologique Propriétés tonneaucomplètedouble (algébrique/topologique)localement convexe réflexif séparable Théorèmes Hahn–Banach Représentation de Riesz graphe fermé principe de délimitation uniforme Kakutani virgule fixeKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Opérateurs adjointlimitécompactHilbert–Schmidtnormalnucléairetraceclasstransposéillimitéunitaire problème de sous-espaceconjecture de MahlerApplicationsespace de Hardythéorie spectrale des équations différentielles ordinairesnoyau de chaleurthéorème d'indexcalcul des variationscalcul fonctionnelopérateur intégralpolynôme de Jonesthéorie des champs quantiques topologiquesgéométrie non commutativehypothèse de Riemanndistribution (ou fonctions généralisées) Sujets avancés propriété d'approximationensemble équilibréThéorie de Choquettopologie faibleDistance de Banach–MazurThéorie de Tomita–Takesaki Catégories: Banach algebrasHardy spacesTheorems in complex analysis