Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
Proof. If f ∈ ker T φ and g ∈ ker T ∗ φ, i.e., P (φf) = 0 and P (φg)=0, then
φf ∈ zH 2 and φg ∈ zH 2 .
Thus φfg, φgf ∈ zH 1 and therefore φfg = 0. If neither f nor g is 0, then by F. and
M. Riesz theorem, φ = 0 a.e. on T, a contradiction.
Corollary 5.1.25. If φ ∈ C(T) then T φ is Fredholm if and only if φ vanishes nowhere.
Proof. By Theorem 5.1.23,
T φ is Fredholm ⇐⇒ π(T φ ) is invertible in T (C(T))/K(H 2 )
⇐⇒ φ is invertible in C(T).
Corollary 5.1.26. If φ ∈ C(T), then σ e (T φ ) = φ(T).
Proof. σ e (T φ ) = σ(T φ + K(H 2 )) = σ(φ) = φ(T).
Theorem 5.1.27. If φ ∈ C(T) is such that T φ is Fredholm, then
index (T φ ) = −wind (φ).
Proof. We claim that if φ and ψ determine homotopic curves in C \ {0}, then
index (T φ ) = index (T ψ ).
To see this, let Φ be a constant map from [0, 1] × T to C \ {0} such that
Φ(0, e it ) = φ(e it ) and Φ(1, e it ) = ψ(e it ).
If we set Φ λ (e it ) = Φ(λ, e it ), then the mapping λ ↦→ T Φλ is norm continuous and
each T Φλ is a Fredholm operator. Since the map index is continuous, index(T φ ) =
index(T ψ ). Now if n = wind(φ) then φ is homotopic in C \ {0} to z n . Since
index (T z n) = −n, we have that index (T φ ) = −n.
Theorem 5.1.28. If U is the unilateral shift on H 2 then comm(U) = {T φ : φ ∈
H ∞ }.
Proof. It is straightforward that UT φ = T φ U for φ ∈ H ∞ , i.e., {T φ : φ ∈ H ∞ } ⊂
comm(U). For the reverse we suppose T ∈ comm(U), i.e., T U = UT . Put φ := T (1).
So φ ∈ H 2 and T (p) = φp for every polynomial p. If f ∈ H 2 , let {p n } be a sequence
of polynomials such that p n → f in H 2 . By passing to a subsequence, we can assume
p n (z) → f(z) a.e. [m]. Thus φp n = T (p n ) → T (f) in H 2 and φp n → φf a.e. [m].
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