Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
Lemma 5.2.8. Let φ = g + f ∈ L ∞ , where f and g are in H 2 . Assume that
f = θ 1 θ 2 a and g = θ 1 b (5.9)
for a ∈ H(θ 1 θ 2 ) and b ∈ H(θ 1 ). Let ψ := θ 1 P H(θ1 )(a) + g. Then T φ is hyponormal if
and only if T ψ is.
Proof. This assertion follows at once from [Gu2, Corollary 3.5].
In view of Lemma 5.2.8, when we study the hyponormality of Toeplitz operators
with bounded type symbols φ, we may assume that the symbol φ = g + f ∈ L ∞ is of
the form
f = θa and g = θb, (5.10)
where θ is an inner function and a, b ∈ H(θ) such that the inner parts of a, b and θ
are coprime.
On the other hand, let f ∈ H ∞ be a rational function. Then we may write
f = p m (z) +
n∑
l∑
i−1
i=1 j=0
a ij
(1 − α i z) li−j (0 < |α i | < 1),
where p m (z) denotes a polynomial of degree m. Let θ be a finite Blaschke product of
the form
∏
n ( ) li
z −
θ = z m αi
.
1 − α i z
Observe that
i=1
a ij
1 − α i z = α ia ij
1 − |α i | 2 ( z − αi
1 − α i z + 1 α i
)
.
Thus f ∈ H(zθ). Letting a := θf, we can see that a ∈ H(zθ) and f = θa. Thus if
φ = g + f ∈ L ∞ , where f and g are rational functions and if T φ is hyponormal, then
we can write
f = θa and g = θb
for a finite Blaschke product θ with θ(0) = 0 and a, b ∈ H(θ).
Now let θ be a finite Blaschke product of degree d. We can write
θ = e iξ
n ∏
i=1
B ni
i , (5.11)
where B i (z) :=
z−αi
1−α , (|α iz i| < 1), n i ≥ 1 and ∑ n
i=1 n i = d. Let θ = e iξ ∏ d
j=1 B j
and each zero of θ be repeated according to its multiplicity. Note that this Blaschke
product is precisely the same Blaschke product in (5.11). Let
ϕ j :=
d j
1 − α j z B j−1B j−2 · · · B 1 (1 ≤ j ≤ d),
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