Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
Proposition 5.1.11. ξ is a contractive ∗-linear mapping from L ∞ to B(H 2 ).
Proof. It is obvious that ξ is contractive and linear. To show that ξ(φ) ∗ = ξ(φ), let
f, g ∈ H 2 . Then
⟨T φ f, g⟩ = ⟨P (φf), g⟩ = ⟨φf, g⟩ = ⟨f, φg⟩ = ⟨f, P (φg)⟩ = ⟨f, T φ g⟩ = ⟨T ∗ φf, g⟩,
so that ξ(φ) ∗ = T ∗ φ = T φ = ξ(φ).
Remark. ξ is not multiplicative. For example, T z T z ≠ I = T 1 = T |z| 2
is not a homomorphism.
In special cases, ξ is multiplicative.
= T zz . Thus ξ
Proposition 5.1.12. T φ T ψ = T φψ
⇐⇒ either ψ or φ is analytic.
Proof. (⇐) Recall that if f ∈ H 2 and ψ ∈ H ∞ then ψf ∈ H 2 . Thus, T ψ f = P (ψf) =
ψf. So
T φ T ψ f = T φ (ψf) = P (φψf) = T φψ f, i.e., T φ T ψ = T φψ .
Taking adjoints reduces the second part to the first part.
(⇒) From a straightforward calculation.
Write M φ for the multiplication operator on L 2 with symbol
(
φ ∈ L ∞ . The
essential range of φ ∈ L ∞ ≡ R(φ) :=the set of all λ for which µ {x : |f(x) − λ| <
)
ϵ} > 0 for any ϵ > 0.
Lemma 5.1.13. If φ ∈ L ∞ (µ) then σ(M φ ) = R(φ).
Proof. If λ /∈ R(φ) then
(
)
∃ ε > 0 such that µ {x : |φ(x) − λ| < ε} = 0, i.e., |φ(x) − λ| ≥ ϵ a.e. [µ].
So
g(x) :=
1
φ(x) − λ ∈ L∞ (X, µ).
Hence M g is the inverse of M φ − λ, i.e., λ /∈ σ(M φ ). For the converse, suppose
λ ∈ R(φ). We will show that
∃ a sequence {g n } of unit vectors ∈ L 2 with the property ||M φ g n − λg n || → 0,
showing that M φ − λ is not bounded below, and hence λ ∈ σ(M φ ). By assumption,
{x ∈ T : |φ(x) − λ| ≤ 1 n
} has a positive measure. So we can find a subset
{
E n ⊆ x ∈ T : |φ(x) − λ| ≤ 1 }
n
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