Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
Theorem 5.2.4 (Nakazi-Takahashi Variation of Cowen’s Theorem). For φ ∈ L ∞ ,
put
E(φ) := {k ∈ H ∞ : ||k|| ∞ ≤ 1 and φ − kφ ∈ H ∞ }.
Then T φ is hyponormal if and only if E(φ) ≠ ∅.
Proof. Let φ = f + g ∈ L ∞ (f, g ∈ H 2 ). By Cowen’s theorem,
T φ is hyponormal ⇐⇒ g = c + T k
f
for some constant c and some k ∈ H ∞ with ||k|| ∞ ≤ 1. If φ = kφ + h (h ∈ H ∞ ) then
φ − kφ = g − kf + f − kg ∈ H ∞ . Thus g − kf ∈ H 2 , so that P (g − kf) = c (c = a
constant), and hence g = c + T k
f for some constant c. Thus T φ is hyponormal. The
argument is reversible.
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