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
everywhere in the unit ball.
Newman showed that the corona theorem can be reduced to an interpolation problem, which was then proved by Carleson.
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 (
Rudin 1991).
Kakutani, Shizuo (1941). "Concrete representation of abstract (M)-spaces. (A characterization of the space of continuous functions.)". Ann. of Math. Series 2. 42 (4): 994–1024.
doi:
10.2307/1968778.
hdl:10338.dmlcz/100940.
JSTOR1968778.
MR0005778.
Koosis, Paul (1980), Introduction to Hp-spaces. With an appendix on Wolff's proof of the corona theorem, London Mathematical Society Lecture Note Series, vol. 40, Cambridge-New York:
Cambridge University Press, pp. xv+376,
ISBN0-521-23159-0,
MR0565451,
Zbl0435.30001