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 Template:Harvtxt and proved by Template:Harvs.

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 Template:Harvtxt, 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

$|f_1|+\cdots +|f_n|\ge \delta$ 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In 1979 Thomas Wolff gave a simplified (but unpublished) proof of the corona theorem, described in and .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 .

See also

• Corona set

References

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