Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
Remark 5.3.11. If T φ
∼ = a weighted shift, what is the form of φ A careful analysis
of the proof of Theorem 5.3.7 shows that
ψ = φ − αφ ∈ H ∞ .
But
⎡
⎤
0 −αa 0
a 0 0 −αa 1
T ψ = T φ − αTφ ∗ =
a 1 0 −αa 2
⎢
.
⎣
a 2 0 ..
⎥
⎦
. .. . ..
⎡
⎤
0 −α
1 0 −α
=
1 0 −α
⎢
.
⎣ 1 0 ..
+ K (K compact)
⎥
⎦
. .. . ..
∼= T z−αz + K.
Thus ran (ψ) = σ e (T ψ ) = σ e (T z−αz ) = ran(z −αz). Thus ψ is a conformal mapping of
D onto the interior of the ellipse with vertices ±i(1+α) and passing through ±(1−α).
On the other hand, ψ = φ − αφ. So αψ = αφ − α 2 φ, which implies
We now have:
φ = 1 (ψ + αψ).
1 − α2 Theorem 5.3.12 (Cowen and Long’s Theorem). For 0 < α < 1, let ψ be a conformal
map of D onto the interior of the ellipse with vertices ±i(1−α) −1 and passing through
±(1 + α) −1 . Then T ψ+αψ
is a subnormal weighted shift that is neither analytic nor
normal.
Proof. Let φ = ψ + αψ. Then φ is a continuous map of D onto D with wind(φ) = 1.
Let
K := 1 − T φ T φ = T φφ − T φ T φ = H ∗ φH φ ,
which is compact since φ is continuous. Now φ − αφ = (1 − α 2 )ψ ∈ H ∞ , so H ψ = 0
and hence, H φ = αH φ . Thus
so that
K = H ∗ φH φ = α 2 H ∗ φH φ = α 2 (1 − T φ T φ ),
KT φ = α 2 (1 − T φ T φ )T φ = α 2 T φ (1 − T φ T φ ) = α 2 T φ K.
By Coburn’s theorem, ker T φ = {0} or ker T φ = {0}. But since
ind(T φ ) = −wind(φ) = −1,
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