Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
Proof. If ∆ ≡ {z ∈ C : |z − 1| < 1} then there exists ϵ > 0 such that ϵ R(φ) ⊆
∆. Hence ||ϵφ − 1|| < 1, which implies ||I − T ϵφ || < 1. Therefore T ϵφ = ϵT φ is
invertible.
Corollary 5.1.21 (Brown-Halmos). If φ ∈ L ∞ , then σ(T φ ) ⊆ conv R(φ).
Proof. It is sufficient to show that every open half-plane containing R(φ) contains
σ(T φ ). This follow at once from Proposition 5.1.20 after a translation and rotation of
the open half-plane to coincide with the open right half-plane.
Proposition 5.1.22. If φ ∈ C(T) and ψ ∈ L ∞ then
Proof. If ψ ∈ L ∞ , f ∈ H 2 then
T φ T ψ − T φψ and T ψ T φ − T ψφ are compact.
T ψ T z f = T ψ P (zf) = T ψ (zf − ̂f(0)z)
= P M ψ
(zf − ̂f(0)z
)
= P (ψzf) − ̂f(0)P (ψz)
= T ψz f − ̂f(0)P (ψz),
which implies that T ψ T z − T ψz is at most a rank one operator. Suppose T ψ T z n − T ψz n
is compact for every ψ ∈ L ∞ and n = 1, · · · , N. Then
T ψ T z N+1 − T ψz N+1 = (T ψ T z N − T ψz N ) T z + (T ψz N T z − T (ψz N )z),
which is compact. Also, since T ψ T z n = T ψz n (n ≥ 0), it follows that T ψ T p − T ψp
is compact for every trigonometric polynomial p. But since the set of trigonometric
polynomials is dense in C(T) and ξ is isometric, we can conclude that T ψ T φ − T ψφ is
compact for ψ ∈ L ∞ and φ ∈ C(T).
Theorem 5.1.23. T (C(T)) contains K(H 2 ) as its commutator and the sequence
0 −→ K(H 2 ) −→ T (C(T)) −→ C(T) −→ 0
is a short exact sequence, i.e., T (C(T))/K(H 2 ) is ∗-isometrically isomorphic to C(T).
Proof. By Proposition 5.1.22 and Corollary 5.1.18.
Proposition 5.1.24. [Co] If φ ≠ 0 a.e. in L ∞ , then
either ker T φ = {0} or ker T ∗ φ = {0}.
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