Corona theorem

In mathematics , the Corona theorem is a theorem from function theory .

sentence

Let the Hardy space , i.e. the Banach algebra of bounded , holomorphic functions ${\displaystyle H^{\infty }}$

${\displaystyle f\colon D\to \mathbb {C} }$

on the circular disc . ${\displaystyle D:=\left\{z\in \mathbb {C} \colon \vert z\vert <1\right\}}$

Be and be so that ${\displaystyle \delta >0}$ ${\displaystyle f_{1},\ldots ,f_{n}\in H^{\infty }}$

${\displaystyle \vert f_{1}(z)\vert +\ldots +\vert f_{n}(z)\vert \geq \delta }$

for everyone . ${\displaystyle z\in D}$

Then there is , so that ${\displaystyle g_{1},\ldots ,g_{n}\in H^{\infty }}$

${\displaystyle f_{1}(z)g_{1}(z)+\ldots +f_{n}(z)g_{n}(z)=1}$

for everyone . ${\displaystyle z\in D}$

Functional analytical interpretation

Let be the set of multiplicative linear functionals on and the set of maximum ideals of Hardy space . Through you have a bijection . ${\displaystyle \Delta }$ ${\displaystyle H^{\infty }}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle H^{\infty }}$ ${\displaystyle \phi \to ker(\phi )}$ ${\displaystyle \Delta \to {\mathcal {M}}}$

Everyone can then have a function through a function or by means of the above bijection ${\displaystyle f\in H^{\infty }}$ ${\displaystyle {\hat {f}}(\phi ):=\phi (f)}$ ${\displaystyle {\hat {f}}\colon \Delta \to \mathbb {C} }$

${\displaystyle {\hat {f}}\colon {\mathcal {M}}\to \mathbb {C} }$

assign. As Gelfand topology is called the weakest topology on , with all ever have. With this topology there is a compact Hausdorff space . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\hat {f}}}$ ${\displaystyle {\mathcal {M}}}$

One can understand as a subspace of by adding the maximum ideal ${\displaystyle D}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle z\in D}$

${\displaystyle M_{z}=\left\{f\in H^{\infty }\colon f(z)=0\right\}}$

assigns.

The Corona theorem is then equivalent to being dense in . ${\displaystyle D}$ ${\displaystyle {\mathcal {M}}}$

history

The Corona theorem in its functional analytical formulation was suggested by Shizuo Kakutani in 1941 and proven by Lennart Carleson in 1962 . The name refers to the fact that the corona is defined by and according to the theorem there is no corona. In 1979 Thomas Wolff gave an elementary proof using - Carleson measures and Littlewood-Paley theory . ${\displaystyle H^{\infty }(D)}$ ${\displaystyle {\mathcal {M}}_{H^{\infty }(D)}\setminus {\overline {D}}}$ ${\displaystyle H^{2}}$

Web links

• Corona theorem (lexicon of mathematics, spectrum of science)
• B. Wick: The Corona Problem