Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
satisfying 0 < µ(E n ) < ∞. Letting g n :=
and hence ||(φ − λ)g n || L 2 ≤ 1 n
−→ 0.
√ χ En
, we have that
µ(En)
∣
∣(φ(x) − λ)g n (x) ∣ ∣ ≤ 1 n |g n(x)|,
Proposition 5.1.14. If φ ∈ L ∞ is such that T φ is invertible, then φ is invertible in
L ∞ .
Proof. In view of Lemma 5.1.13, it suffices to show that
If T φ is invertible then
So for n ∈ Z and f ∈ H 2 ,
T φ is invertible =⇒ M φ is invertible.
∃ ε > 0 such that ||T φ f|| ≥ ε||f||, ∀f ∈ H 2 .
||M φ (z n f)|| = ||φz n f|| = ||φf|| ≥ ||P (φf)|| = ||T φ f|| ≥ ε||f|| = ε||z n f||.
Since {z n f : f ∈ H 2 , n ∈ Z} is dense in L 2 , it follows ||M φ g|| ≥ ε||g|| for g ∈
L 2 . Similarly, ||M φ f|| ≥ ε||f|| since T ∗ φ = T φ is also invertible. Therefore M φ is
invertible.
Theorem 5.1.15 (Hartman-Wintner). If φ ∈ L ∞ then
(i) R(φ) = σ(M φ ) ⊂ σ(T φ )
(ii) ||T φ || = ||φ|| ∞ (i.e., ξ is an isometry).
Proof. (i) From Lemma 5.1.13 and Proposition 5.1.14.
(ii) ||φ|| ∞ = sup λ∈R(φ) |λ| ≤ sup λ∈σ(Tφ)|λ| = r(T φ ) ≤ ||T φ || ≤ ||φ|| ∞ .
From Theorem 5.1.15 we can see that
(i) If T φ is quasinilpotent then T φ = 0 because R(φ) ⊆ σ(T φ ) = {0} ⇒ φ = 0.
(ii) If T φ is self-adjoint then φ is real-valued because R(φ) ⊆ σ(T φ ) ⊆ R.
If S ⊆ L ∞ , write T (S) := the smallest closed subalgebra of L(H 2 ) containing
{T φ : φ ∈ S}.
If A is a C ∗ -algebra then its commutator ideal C is the closed ideal generated by
the commutators [a, b] := ab − ba (a, b ∈ A). In particular, C is the smallest closed
ideal in A such that A/C is abelian.
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