Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
Example 5.3.16. Let ψ be the ellipse map as in the example of Cowen and Long.
Then
T ψ
∼ = Tφ with φ = i e− iθ 2 (1 + α 2 e iθ (
)
1 − α 2 ψ +
Proof. Note that
Thus we have
)
αeiθ + α
1 + α 2 e iθ ψ
T ∼ = e iθ 2 T and T + λT
∗ ∼ = e
iθ
2 T + λe
− iθ 2 T ∗ .
T ψ
∼ = (1 − α 2 ) 1 2 (T − αT ∗ )
∼= (1 − α 2 ) 1 2 i(T + αT ∗ )
∼= (1 − α 2 ) 1 2 ie
− iθ 2 (T + αe iθ T ∗ )
(
)
∼= (1 − α 2 ) −1 ie − iθ 2 T ψ + αT ψ
+ αe iθ (T ψ
+ αT ψ )
∼= (1 − α 2 ) −1 i e − iθ 2 T(1+α 2 e iθ )ψ+α(1+e iθ )ψ
(−π < θ < π)
∼= i e− iθ 2 (1 + α 2 e iθ )
1 − α 2 T ψ+
αe iθ +α
1+α 2 e iθ ψ (−π < θ ≤ π).
(−π < θ ≤ π)
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