Corona theorem
E943111
The Corona theorem is a fundamental result in complex analysis that characterizes when bounded analytic functions on the unit disk can be solved in a certain type of division problem, showing that the maximal ideal space of the disk algebra has no "corona."
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in complex analysis ⓘ |
| appliesTo |
algebra H^∞(D)
ⓘ
disk algebra A(D) ⓘ |
| author | Lennart Carleson NERFINISHED ⓘ |
| concerns |
corona problem
ⓘ
maximal ideal space of the disk algebra ⓘ |
| conclusion | existence of bounded analytic g_1,...,g_n with f_1 g_1+...+f_n g_n=1 ⓘ |
| coreCondition | bounded analytic functions f_1,...,f_n with no common zero on the unit disk ⓘ |
| difficulty | proof is technically complex ⓘ |
| domain | open unit disk in the complex plane ⓘ |
| field |
complex analysis
ⓘ
functional analysis ⓘ operator theory ⓘ |
| generalizationOf | classical division problems in function algebras ⓘ |
| hasAlternativeProof |
methods using Carleson measure estimates
ⓘ
methods using \\bar{\partial}-techniques ⓘ methods using functional-analytic techniques ⓘ |
| hasGeneralization |
corona theorems for several complex variables
ⓘ
corona theorems on Riemann surfaces ⓘ corona theorems on strictly pseudoconvex domains ⓘ operator corona theorem ⓘ |
| implies |
maximal ideal space of H^∞(D) is connected to the unit disk
ⓘ
no additional maximal ideals lying over the boundary of the unit disk ⓘ |
| influenced |
Hardy space theory
ⓘ
development of function algebra theory ⓘ operator-valued function theory ⓘ |
| isCornerstoneOf | corona theory in function algebras ⓘ |
| mainObject |
bounded analytic functions
ⓘ
disk algebra ⓘ unit disk ⓘ |
| namedAfter | corona of the maximal ideal space ⓘ |
| publishedIn | Acta Mathematica NERFINISHED ⓘ |
| relatedOpenProblem |
Carleson’s corona problem in higher dimensions
NERFINISHED
ⓘ
structure of maximal ideal space of H^∞(D) ⓘ |
| relatedTo |
Beurling’s theorem
NERFINISHED
ⓘ
Carleson measures ⓘ Nevanlinna–Pick interpolation NERFINISHED ⓘ interpolation in H^∞ ⓘ |
| statementAbout |
absence of corona in maximal ideal space of disk algebra
ⓘ
solvability of certain division problems in H^∞(D) ⓘ |
| surveyedIn |
monographs on H^∞ spaces
ⓘ
“The corona theorem” by Thomas W. Gamelin NERFINISHED ⓘ |
| usesConcept |
Banach algebras
NERFINISHED
ⓘ
Gelfand transform NERFINISHED ⓘ bounded holomorphic functions ⓘ maximal ideals ⓘ |
| yearProved | 1962 ⓘ |
Referenced by (1)
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