The Corona Problem in the Dirichlet Space Brett D. Wick Georgia Institute of Technology School of Mathematics MSRI Summer Graduate Program: The Dirichlet Space: Connections between Operator Theory, Function Theory, and Complex Analysis June 27, 2011 to July 1, 2011 B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 1 / 131 Lecture Outlines & Topics Covered Motivations for the Problem ∞ The Corona Problem for H (D) Carleson measures and ∂-problems; Wolff’s proof of the Corona Problem; Jones’ constructive solution to ∂b = µ; The Corona Problem for MD Xiao’s Theorem on the Dirichlet space The Corona Problem for Multiplier Algebras with the Complete Nevanlinna-Pick Property Reproducing kernel Hilbert function spaces with Complete Nevanlinna-Pick kernel; The Baby Corona Problem & The Corona Problem; Toeplitz Corona Theorem; The Corona Problem in Several Variables B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 2 / 131 Motivations for the Problem Motivations for the Problem B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 3 / 131 Motivations for the Problem Where Did the Name Come From? The Beer Problem? B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 4 / 131 Motivations for the Problem Commutative Banach Algebras A (commutative) Banach algebra A is a complex (commutative) algebra A that is also a Banach space under a norm that satisfies kfgkA ≤ kf kA kgkA f , g ∈ A. We will also assume that there is an identity element 1 ∈ A and that our algebra is commutative. An element f ∈ A is invertible if there exists an element g ∈ A such that fg = 1 and write f −1 for g. We let A−1 = {f ∈ A : f −1 exists}. Finally, we need to consider the non-zero multiplicative linear functionals on the algebra A. These are simply the complex homomorphisms m : A → C. Note that we trivially have that m(1) = 1. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 5 / 131 Motivations for the Problem Commutative Banach Algebras Lemma 1 Every complex homomorphism from A to C is a continuous linear functional with norm at most 1. Namely, kmk = sup |m(f )| ≤ 1. f ∈A,kf kA≤1 Proof: If m is unbounded or if kmk > 1 then we can find an element f ∈ A with kf kA < 1 but m(f ) = 1. Since kf kA < 1 we have ∞ ∞ ∞ 1 X f n ≤ X kf nk ≤ X kf kn = . A A 1 − kf k n=0 A n=0 n=0 A P∞ n −1 Note that we have (1 − f ) n=0 f = 1, so 1 − f ∈ A . However, 1 = m(1) = m((1 − f )(1 − f )−1) = m((1 − f )−1)(m(1) − m(f )) = 0. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 6 / 131 Motivations for the Problem Commutative Banach Algebras Lemma 2 Suppose that M is a maximal ideal of A. Then M is the kernel of a multiplicative linear functional m : A → C. Conversely, suppose that m : A → C is a multiplicative linear functional. Then ker m is a maximal ideal. Proof: For the converse, it is immediate that the ker m is an ideal. The maximality follows since the codimension of any linear functional is 1. Namely, dim(A\ ker m) = 1. In the other direction, one first shows that the maximal ideal M is closed. Then one shows that the quotient algebra B = A/M satisfies B = C1 where 1 = 1 + M denotes the unit in the quotient algebra. The quotient mapping will then define the multiplicative linear functional, and the kernel of this mapping with then be M. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 7 / 131 Motivations for the Problem Commutative Banach Algebras Let MA denote the set of complex homomorphisms of A (called the maximal ideal space of the Banach algebra). ∗ By Lemma1, we have that MA ⊂ Ball1 (A ). ∗ Endow MA with the weak-∗ topology of A (called the Gelfand topology). Namely, the basic neighborhood of a m0 ∈ MA is determined by > 0 and by elements f1,..., fn ∈ A such that ∗ V = {m ∈ A : kmk ≤ 1, |m(fj ) − m0(fj )| < , 1 ≤ j ≤ n} . Because ∗ MA = {m ∈ A : kmk ≤ 1, m(fg) = m(f )m(g), f , g ∈ A} ∗ we have that MA is a weak-∗ closed subset of the unit ball of A . The Banach-Alaoglu Theorem gives that the ball of A∗ is weak-∗ compact, which implies that MA is a compact Hausdorff space. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 8 / 131 Motivations for the Problem Where Did the Name Come From? ∞ The Corona Problem for H (D) ∞ The Banach algebra H (D) is the collection of all analytic functions on the disc such that kf k ∞ := sup |f (z)| < ∞. H (D) z∈D ∞ Let ϕ : H (D) → C be a non-zero multiplicative linear functional. Namely, ϕ(fg) = ϕ(f )ϕ(g) and ϕ(f + g) = ϕ(f ) + ϕ(g). As we have seen for any multiplicative linear functional sup |ϕ(f )| ≤ kf kH∞( ) . ∞ D f ∈H (D) To each z ∈ D we can associate a multiplicative linear functional on ∞ H (D): ϕz (f ) := f (z) (point evaluation at z). B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 9 / 131 Motivations for the Problem Where Did the Name Come From? ∞ The Corona Problem for H (D) Every multiplicative linear functional ϕ determines a maximal (proper) ∞ ∞ ideal of H (D): ker ϕ = {f ∈ H (D): ϕ(f ) = 0}. ∞ Conversely, if M is a maximal (proper) ideal of H (D) then M = ker ϕ for some multiplicative linear functional. ∞ The maximal ideal space of H ( ), M ∞ , is the collection of all D H (D) multiplicative linear functionals ϕ. We then have that the maximal ideal space is contained in the unit ball of ∞ the dual space H (D). If we put the weak-∗ topology on this space then M ∞ is a compact Hausdorff space. H (D) The proceeding discussion then shows that ⊂ M ∞ . D H (D) B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 10 / 131 Motivations for the Problem Where Did the Name Come From? ∞ The Corona Problem for H (D) ∞ One then defines the Corona of H ( ) to be M ∞ \ . D H (D) D In 1941, Kakutani asked if there was a Corona in the maximal ideal space ∞ M ∞ of H ( ), i.e. whether or not the disc was dense in M ∞ ? H (D) D D H (D) B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 11 / 131 Motivations for the Problem Where Did the Name Come From? ∞ The Corona Problem for H (D) Using basic functional analysis, Kakutani’s question can be phrased as the following question about analytic functions on the unit disc: Theorem 1 The open disc D is dense in MH∞ if and only if the following condition ∞ holds: If f1,..., fn ∈ H (D) and if max |fj (z)| ≥ δ > 0 1≤j≤n ∞ then there exists g1, ··· , gn ∈ H (D) such that f1g1 + ··· + fngn = 1. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 12 / 131 Motivations for the Problem Proof of Theorem1 Proof: Suppose that D is dense in MH∞ . Then, by continuity we have that max |m(fj )| ≥ δ 1≤j≤n for all m ∈ MH∞ . This implies that {f1,..., fn} is in no proper ideal of ∞ H (D). Hence the ideal generated by {f1,..., fn} must contain the ∞ constant function 1 and so there exists g1,..., gn ∈ H (D) such that 1 = f1g1 + ··· + fngn. Conversely, suppose that D is not dense in MH∞ . Then for some m0 ∈ MH∞ has a neighborhood disjoint from D and this neighborhood has the form n V = ∩j=1{m : |m(fj )| < δ} ∞ where δ > 0 and f1,..., fn ∈ H (D) with m0(fj ) = 0. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 13 / 131 Motivations for the Problem Proof of Theorem1, Continued Since D ∩ V = ∅ we have that max |fj (z)| ≥ δ > 0. 1≤j≤n But, it is not possible that we have 1 = f1g1 + ··· + fngn since we have that m0(fj ) = 0 for all 1 ≤ j ≤ n. Exercise 2 Let A( ) = {f ∈ Hol( ) ∩ C( ): sup |f (z)| < ∞} denote the disc D D D z∈D algebra. Show that M = . A(D) D B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 14 / 131 Motivations for the Problem Where Did the Name Come From? ∞ The Corona Problem for H (D) Kakutani’s question was settled in 1962 by Carleson: = M ∞ . D H (D) Theorem 3 (Carleson’s Corona Theorem) N ∞ Let {fj }j=1 ∈ H (D) satisfy N X 2 0 < δ ≤ |fj (z)| ≤ 1, ∀z ∈ D. j=1 Then there are functions {g }N in H∞ ( ) with Lennart Carleson j j=1 D N X fj (z) gj (z) = 1 ∀z ∈ D and kgj k∞ ≤ Cδ,N . j=1 B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 15 / 131 Motivations for the Problem Reasons to Care about the Corona Problem 2 2 1 Angles between Invariant Subspaces: Kθ := H (D) θH (D) The angle between Kθ1 and Kθ2 is positive if and only if there exists ψ1 and ψ2 such that ψ1θ1 + ψ2θ2 = 1. ∞ 2 Interpolation in H (D): Let B(z) be a Blaschke product with zeros ∞ {zk } = Z, and let f ∈ H (D) be such that |f (z)| + |B(z)| ≥ δ > 0 ∀z ∈ D. ∞ Then there exists p, q ∈ H (D) such that pf + qB = 1 if and only if there is a solution to the interpolation problem 1 p(zk ) = ∀zk ∈ Z. f (zk ) 3 Control Theory B.