Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
Theorem 5.1.9 (Inner-Outer Factorization). If f is a nonzero function in H 2 , then
∃ an inner function ϕ and an outer function g in H 2
such that f = ϕg.
In particular, if f ∈ H ∞ , then g ∈ H ∞ .
Proof. Observe M ≡ ∨ {z n f : n ≥ 0} ∈ Lat U. By Beurling’s theorem,
∃ an inner function ϕ s.t. M = ϕH 2 .
Let g ∈ H 2 be such that f = ϕg. We want to show that g is outer. Put N ≡ ∨ {z n g :
n ≥ 0}. Again there exists an inner function ψ such that N = ψH 2 . Note that
ϕH 2 := ∨ {z n f : n ≥ 0} = ∨ {z n ϕg : n ≥ 0} = ϕψH 2 .
Therefore there exists a function h ∈ H 2 such that ϕ = ϕψh so that ψ = h ∈ H 2 .
Hence ψ is a constant by Corollary 5.1.7. So N = H 2 and g is outer. Assume f ∈ H ∞
with f = ϕg. Thus |g| = |f| a.e. on ∂D, so that g must be bounded, i.e., g ∈ H ∞ .
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