Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
Theorem 5.1.16. If C is the commutator ideal in T (L ∞ ), then the mapping ξ c
induced from L ∞ to T (L ∞ )/C by ξ is a ∗-isometrical isomorphism. Thus there is a
short exact sequence
Proof. See [Do1].
0 −→ C −→ T (L ∞ ) −→ L ∞ −→ 0.
The commutator ideal C contains compact operators.
Proposition 5.1.17. The commutator ideal in T (C(T)) = K(H 2 ). Hence the commutator
ideal of T (L ∞ ) contains K(H 2 ).
Proof. Since T z is the unilateral shift, we can see that the commutator ideal of
T (C(T)) contains the rank one operator Tz ∗ T z − T z Tz ∗ . Moreover, T (C(T)) is irreducible
since T z has no proper reducing subspaces by Beurling’s theorem. Therefore
T (C(T)) contains K(H 2 ). Since T z is normal modulo a compact operator and generates
the algebra T (C(T)), it follows that T (C(T))/K(H 2 ) is commutative. Hence
K(H 2 ) contains the commutator ideal of T (C(T)). But since K(H 2 ) is simple (i.e., it
has no nontrivial closed ideal), we can conclude that K(H 2 ) is the commutator ideal
of T (C(T)).
Corollary 5.1.18. There exists a ∗-homomorphism ζ : T (L ∞ )/K(H 2 ) −→ L ∞
such that the following diagram commutes:
T (L ∞ )
❏ ❏❏❏❏❏❏❏❏
ρ
π
ζ
♣♣♣♣♣♣♣♣♣♣♣
L ∞ (T)
T (L ∞ )/K(H 2 )
Corollary 5.1.19. Let φ ∈ L ∞ . If T φ is Fredholm then φ is invertible in L ∞ .
Proof. If T φ is Fredholm then π(T φ ) is invertible in T (L ∞ )/K(H 2 ), so φ = ρ(T φ ) =
(ζ ◦ π)(T φ ) is invertible in L ∞ .
From Corollary 5.1.18, we have:
(i) ||T φ || ≤ ||T φ + K|| for every compact operator K because ||T φ || = ||φ|| ∞ =
||ζ(T φ + K)|| ≤ ||T φ + K||.
(ii) The only compact Toeplitz operator is 0 because ||K|| ≤ ||K + K|| ⇒ K = 0.
Proposition 5.1.20. If φ is invertible in L ∞ such that R(φ) ⊆ the open right halfplane,
then T φ is invertible.
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