Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
5.1.2 Hardy Spaces
If f ∈ H 2 and f(z) = ∑ ∞
n=0 a nz n is its Fourier series expansion, this series converges
uniformly on compact subsets of D. Indeed, if |z| ≤ r < 1, then
(
∞∑
∑ ∞
|a n z n | ≤
n=m
n=m
|a n | 2 ) 1
2 ( ∞ ∑
n=m
|z| 2n ) 1
2
≤ ||f|| 2
( ∞
∑
n=m
r 2n ) 1
2
.
Therefore it is possible to identify H 2 with the space of analytic functions on the unit
disk whose Taylor coefficients are square summable.
Proposition 5.1.6. If f is a real-valued function in H 1 then f is constant.
Proof. Let α = ∫ fdm. By hypothesis, we have α ∈ R. Since f ∈ H 1 , we have
∫
fz n dm = 0 for n ≥ 1. So ∫ (f − α)z n dm = 0 for n ≥ 0. Also,
∫
0 =
∫
(f − α)z n dm =
(f − α)z −n dm (n ≥ 0),
so that ∫ (f − α)z n dm = 0 for all integers n. Thus f − α annihilates all the trigonometric
polynomials. Therefore, f − α = 0 in L 1 .
Corollary 5.1.7. If ϕ is inner such that ϕ = 1 ϕ ∈ H2 then ϕ is constant.
Proof. By hypothesis, ϕ+ϕ and ϕ−ϕ
i
5.1.6, they are constant, so is ϕ.
are real-valued functions in H 2 . By Proposition
The proof of the following important theorem uses Beurling’s theorem.
Theorem ( 5.1.8 (The F. and)
M. Riesz Theorem). If f is a nonzero function in H 2 ,
then m {z ∈ ∂D : f(z) = 0} = 0. Hence, in particular, if f, g ∈ H 2 and if fg = 0
a.e. then f = 0 a.e. or g = 0 a.e.
Proof. Let △ be a Borel set of ∂D and put
M := {h ∈ H 2 : h(z) = 0 a.e. on △}.
Then M is an invariant subspace for the unilateral shift. By Beurling’s theorem,
if M ̸= {0}, then there exists an inner function ϕ such that M = ϕH 2 . Since
ϕ = ϕ · 1 ∈ M, it follows ϕ = 0 on △. But |ϕ| = 1 a.e., and hence M = {0}.
A function f in H 2 is called an outer function if
H 2 = ∨ {z n f : n ≥ 0}.
So f is outer if and only if it is a cyclic vector for the unilateral shift.
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