Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
If the answer to Problem 5.5 is affirmative, i.e., the Cowen’s remark is true then
for φ = g + f,
T φ is subnormal =⇒ g − λf ∈ H 2
with |λ| < 1 =⇒ g = λf + c (c a constant),
which says that the answer to Problem 5.3 is negative.
When ψ is as in Theorem 5.3.12, we examine the question: For which λ, is T ψ+λψ
subnormal
We then have:
Theorem 5.3.14. Let λ ∈ C and 0 < α < 1. Let ψ be the conformal map of the disk
onto the interior of the ellipse with vertices ±(1 + α)i passing through ±(1 − α). For
φ = ψ + λψ, T φ is subnormal if and only if λ = α or λ = αk e iθ +α
1+α k+1 e iθ (−π < θ ≤ π).
To prove Theorem 5.3.14, we need an auxiliary lemma:
Proposition 5.3.15. Let T be the weighted shift with weights
w 2 n =
n∑
α 2j .
Then T + µT ∗ is subnormal if and only if µ = 0 or |µ| = α k (k = 0, 1, 2, · · · ).
Proof. See [CoL].
j=0
Proof of Theorem 5.3.14. By Theorem 5.3.12, T ψ+αψ
∼ = (1 − α 2 ) 3 2 T , where T is a
weighted shift of Proposition 5.3.15. Thus T ψ
∼ = (1 − α 2 ) 1 2 (T − αT ∗ ), so
(
T φ = T ψ + λTψ ∗ ∼ = (1 − α 2 ) 1 2 (1 − λα) T + λ − α )
1 − λα T ∗ .
Applying Proposition 5.3.15 with λ−α
1−λα
in place of µ gives that for k = 0, 1, 2, · · · ,
λ − α
∣1 − λα∣ = αk ⇐⇒ λ − α
1 − λα = αk e iθ
⇐⇒ λ − α = α k e iθ − λα k+1 e iθ
⇐⇒ λ(1 + α k+1 e iθ ) = α + α k e iθ
⇐⇒ λ =
α + αk e iθ
1 + α k+1 e iθ (−π < θ ≤ π) □
However we find that, surprisingly, some analytic Toeplitz operators are unitarily
equivalent to some non-analytic Toeplitz operators. So C. Cowen noted that subnormality
of Toeplitz operators may not be the wrong question to be studying.
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