mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the corona theorem is a result about the

spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

of the bounded

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...

s on the open unit disc, conjectured by and proved by . The commutative

Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...

and

Hardy space In complex analysis, the Hardy spaces (or Hardy classes) H^p are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real anal ...

''H''^∞ consists of the bounded

s on the open unit disc ''D''. Its

''S'' (the closed

maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...

s) contains ''D'' as an open subspace because for each ''z'' in ''D'' there is a

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 In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...

and ''D'' is not. The complement of the closure of ''D'' in ''S'' was called the corona by , 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 ''f''₁,...,''f''_''n'' generate the unit ideal of ''H''^∞ if and only if there is some δ>0 such that :

, f_1, +\cdots+, f_n, \ge\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. In 1979

Thomas Wolff Thomas Hartwig Wolff (July 14, 1954, New York City – July 31, 2000, Kern County) was an American mathematician, working primarily in the fields of harmonic analysis, complex analysis, and partial differential equations. As an undergraduate ...

gave a simplified (but unpublished) proof of the corona theorem, described in and . Cole later showed that this result cannot be extended to all open Riemann surfaces . As a by-product, of Carleson's work, the Carleson measure was introduced 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 the corona theorem, then the conclusion follows easily from the theory of commutative Banach algebra .

References

* * * * * * * . * . {{Functional analysis Banach algebras Hardy spaces Theorems in complex analysis

See also

References