Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
Proof. Let φ = f + g (f, g ∈ H 2 ). For every polynomial p ∈ H 2 ,
⟨(T ∗ φT φ − T φ T ∗ φ)p, p⟩ = ⟨T φ p, T φ p⟩ − ⟨T ∗ φp, T ∗ φp⟩
Since polynomials are dense in H 2 ,
= ⟨fp + P gp, fp + P gp⟩ − ⟨P fp + gp, P fp + gp⟩
= ⟨fp, fp⟩ − ⟨P fp, P fp⟩ − ⟨gp, gp⟩ + ⟨P gp, P gp⟩
= ⟨fp, (I − P )fp⟩ − ⟨gp, (I − p)gp⟩
= ⟨(I − P )fp, (I − P )fp⟩ − ⟨(I − P )gp, (I − P )gp⟩
= ||H f
p|| 2 − ||H g p|| 2 .
T φ hyponormal ⇐⇒ ||H g u|| ≤ ||H f
u||, ∀u ∈ H 2 (5.2)
Write K := cl ran(H f
) and let S be the compression of the unilateral shift U to K.
Since K is invariant for U ∗ (why: H f
U = U ∗ H f
), we have S ∗ = U ∗ | K . Suppose T φ is
hyponormal. Define A on ran(H f
) by
Then A is well defined because by (5.3)
A(H f
u) = H g u. (5.3)
H f
u 1 = H f
u 2 =⇒ H f
(u 1 − u 2 ) = 0 =⇒ H g (u 1 − u 2 ) = 0.
By (5.2), ||A|| ≤ 1, so A has an extension to K, which will also be denoted A. Observe
that
H g U = AH f
U = AU ∗ H f
= AS ∗ H f
and H g U = U ∗ H g = U ∗ AH f
= S ∗ AH f
.
Thus AS ∗ = S ∗ A on K since ranH f
is dense in K, and hence SA ∗ = A ∗ S. By
Sarason’s interpolation theorem,
∃ k ∈ H ∞ (D) with ||k|| ∞ = ||A ∗ || = ||A|| s.t. A ∗ = the compression of T k to K.
Since Tk ∗H f = H f T k ∗, we have that K is invariant for T k ∗ = T k
, which means that A
is the compression of T k
to K and
H g = T k
H f
(by (5.3)). (5.4)
Conversely, if (5.4) holds for some k ∈ H ∞ (D) with ||k|| ∞ ≤ 1, then (5.2) holds for
all u, and hence T φ is hyponormal. Consequently,
T φ hyponormal ⇐⇒ H g = T k
H f
.
But H g = T k
H f
if and only if ∀ u, v ∈ H ∞ ,
⟨zuv, g⟩ = ⟨H g u, v ∗ ⟩ = ⟨T k
H f
u, v ∗ ⟩ = ⟨H f
u, kv ∗ ⟩
= ⟨zuk ∗ v, f⟩ = ⟨zuv, k ∗ f⟩ = ⟨zuv, T k ∗f⟩.
Since ∨ {zuv : u, v ∈ H ∞ } = zH 2 , it follows that
H g = T k
H f
⇐⇒ g = c + T h
f for h = k ∗ .
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