For the special case of E =D the AAK theorem effectively states that
sk(f)<∞ ⇔ there exists a Blaschke product b∈Bk such thatf b∈H∞ and then sk(f) =kf bk∞.
As a first step to proving the theorem we will now prove this. To do so we first need the following general result.
Lemma 7.3.1 Let K :X×X →C be a positive definite kernel on a set X. Then
any functionφthat is a bounded multiplier of a dense subspaceSof H(K)back into
Proof: Assume, without loss of generality, that kMφ|Sk = 1 and let f be any function in H(K), with normr say. SinceS is a dense subspace of H(K) we can find functions fi ∈S, i∈ N, with the same normr, such that fi →f in norm and hence also weakly (i.e. pointwise). Then
kMφ|Sk= 1 ⇒ kφfik ≤r for all i
⇒ r2K−(φfi)(φfi)∗ ≥0 for all i
⇒ r2K−(φf)(φf)∗ ≥0 where the last implication holds becauser2K−(φf
i)(φfi)∗ →r2K−(φf)(φf)∗ pointwise and the pointwise limit of positive kernels must also be positive. Therefore φf ∈ H(K) and kφfk ≤ r = kfk, so φ indeed multiplies all functions in H(K) contractively back into H(K).
We can now give function-theoretic meaning to the singular values of functions on the disc.
Lemma 7.3.2 Let f :D→C be any given function and K be the Szeg¨o kernel. If
there exists a Blaschke product b∈Bk such that f b∈H∞ then sk(f) =kf bk∞ = lim
r→1−supz∈T|f(rz)|.
Otherwise sk(f) =∞.
Proof: We first prove that ifb ∈Bk and f b∈H∞ then sk(f)≤ kf bk∞:
If such a Blaschke product b exists, then Mf b maps H2 into H2 with norm
kf bk∞, so Mf maps bH2 into H2 with the same norm, since multiplication by any Blaschke product is an isometry on H2. But bH2 has codimension
in H2 equal to the number of Blaschke factors in b, i.e. no more than k, so
Secondly, we show that if sk(f) < ∞ then there exists b ∈ Bk such that f b∈H∞ and kf bk∞≤sk(f):
Because scaling a function simply scales its singular values, we may assume that sk(f) = 1. Consider the linear manifold H = {h ∈ H2 : f h ∈ H2} of functions in H2 that M
f maps back into H2. H is clearly invariant under multiplication by z and moreover so is its closure in H2, since if h
i ∈H and hi →h∈H2 in norm, then zhi →zhin norm andzhi ∈H, sozh∈clos(H). Therefore, by Beurling’s theorem [Beu49], clos(H) = bH2 for some inner
function b. Since sk(f) < ∞ then, by the definition of sk(f), H must have codimension ≤k inH2, so b must be a finite Blaschke product with at most k factors.
Now consider the unitary equivalence Mb :H2 →bH2.
H ⊆ bH2 = clos(H) 6 Mb 6 Mb Mb−1H ⊆ H2
Since H is dense in bH2 then, by this equivalence, M−1
b H is dense in H2, so multiplication by f b contractively maps a dense subspace ofH2, i.e. M−1
b H, back into H2. By lemma 7.3.1, f bis therefore a contractive multiplier on the
whole of H2, so f b∈H∞ and kf bk
∞≤1, as claimed.
Finally note that, because Blaschke products are unimodular around the disc edge, kf bk∞ = limr→1−supz∈T|f(rz)| for any Blaschke product b such that
f b∈H∞. The lemma now follows from the fact that this value is independent of the Blaschke product b chosen, together with the above two implications.
Corollary 7.3.3 Relative to the Szeg¨o kernel, the singular values of any function
f :D→C are given by
(s0(f), . . . , sk−1(f), sk(f), sk+1(f), . . .) = (∞, . . . ,∞, s, s, . . .)
where s = limr→1−supz∈T|f(rz)| and k is the smallest integer such that f b ∈H
∞
for some b ∈Bk.
Taking φ =f b in this corollary completes our proof that the AAK theorem holds when E =D. However, the purpose of proving this has been mainly to enable us
to see that the AAK theorem can be viewed as a function extension problem. Starting from the function f with k’th singular value sk(f), suppose we could extend f from E to D without increasing the k’th singular value. The resulting
functionψ :D→Cwould satisfysk(ψ) = sk(f) and so by lemma 7.3.2 there would exist a Blaschke product b ∈ Bk such that ψb ∈ H∞ and kψbk∞ = sk(f). The functionφ =ψbwould therefore satisfyφ∈H∞, kφk∞=sk(f) andφ|E =ψb|E =
f b|E and so satisfy the requirements of the theorem.
It appears, therefore, that proving the AAK theorem is now a matter of showing that a function can always be extended from E to D without increasing its k’th
singular value. Our approach will therefore be as with Pick’s theorem in chapter 1. The main issue involved is whether it is always possible to one-point extend a function without increasing sk. The next section tackles this problem.