“Corona theorem”的意思、由来-开放百科全书

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Corona theorem

释义

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In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by {{harvtxt|Kakutani|1941}} and proved by {{harvs|authorlink=Lennart Carleson|first=Lennart|last=Carleson|year=1962|txt=yes}}.

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 {{harvtxt|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,...,f_n generate the unit ideal of H^∞ if and only if there is some δ>0 such that

everywhere in the unit ball.

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

In 1979 Thomas Wolff gave a simplified (but unpublished) proof of the corona theorem, described in {{harv|Koosis|1980}} and {{harv|Gamelin|1980}}.

Cole later showed that this result cannot be extended to all open Riemann surfaces {{harv|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 assume the continuity up to the boundary in Corona's theorem, then the conclusion follows easily from the theory of Commutative Banach algebra {{harv|Rudin}}.

References

{{citation|mr=0141789 | zbl = 0112.29702

|last= Carleson
|first= Lennart
|author-link= Lennart Carleson
|title=Interpolations by bounded analytic functions and the corona problem
|journal= Annals of Mathematics
|issue= 3
|volume=76
|year= 1962
|pages=547–559
|doi=10.2307/1970375
|jstor=1970375
}}

{{citation|mr=0521440 | zbl = 0418.46042

|last=Gamelin|first= T. W.
|title=Uniform algebras and Jensen measures.
|series=London Mathematical Society Lecture Note Series
|volume= 32
|publisher= Cambridge University Press
|place= Cambridge-New York
|year= 1978
|pages= iii+162
|isbn= 978-0-521-22280-8}}

{{citation|mr=0599306 | zbl = 0466.46050

|last=Gamelin|first= T. W.
|title=Wolff's proof of the corona theorem
|journal=Israel Journal of Mathematics
|volume= 37
|year=1980
|issue= 1–2
|pages= 113–119
|doi=10.1007/BF02762872}}

{{citation|mr=0565451 | zbl = 0435.30001

|last=Koosis|first= Paul
|title=Introduction to H^p-spaces. With an appendix on Wolff's proof of the corona theorem
|series=London Mathematical Society Lecture Note Series
|volume= 40
|publisher= Cambridge University Press
|place= Cambridge-New York
|year=1980
|pages= xv+376
|isbn= 0-521-23159-0}}

{{citation|mr=0106290 | zbl = 0092.11802

|last= Newman
|first= D. J.
|title= Some remarks on the maximal ideal structure of H^∞
|journal= Annals of Mathematics
|issue= 2
|volume= 70
|year= 1959
|pages= 438–445
|doi=10.2307/1970324
|jstor=1970324
}}

{{citation

|last=Rudin
|first= Walter
|title=Functional Analysis
|year=1991
|pages=279}}.

{{citation

|mr=0125442 | zbl = 0139.30402
|last=Schark
|first= I. J.
|title=Maximal ideals in an algebra of bounded analytic functions
|journal=Journal of Mathematics and Mechanics
|volume= 10
|url=http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1961/10/10050
|year=1961
|pages=735–746}}.

3 : Banach algebras|Hardy spaces|Theorems in complex analysis

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