Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
Corollary 5.3.20. If T φ is a 2-hyponormal operator such that E(φ) contains at least
two elements then T φ is normal or analytic, so that T φ is subnormal.
Proof. This follows from Corollary 5.3.19 and the fact ([NaT, Proposition 8]) that if
E(φ) contains at least two elements then φ is of bounded type.
From Corollaries 5.3.19 and 5.3.20, we can see that if T φ is 2-hyponormal but not
subnormal then φ is not of bounded type and E(φ) consists of exactly one element.
For a strategy to answer Problem 5.7 we will introduce the notion of “weak subnormality,”
which was introduced by R. Curto and W.Y. <strong>Lee</strong> [CuL2]. Recall that
the operator [ ] T is subnormal if and only if there exist operators A and B such that
T A
̂T := is normal, i.e.,
0 B
⎧
⎪⎨ [T ∗ , T ] := T ∗ T − T T ∗ = AA ∗
A
⎪⎩
∗ T = BA ∗
[B ∗ , B] + A ∗ A = 0.
(5.33)
We now introduce:
Definition 5.3.21. [CuL2] An operator T ∈ B(H) is said to be weakly subnormal if
there exist operators A ∈ L(H ′ , H) and B ∈ L(H ′ ) such that the first two conditions
in (5.33) hold: [T ∗ , T ] = AA ∗ and A ∗ T = BA ∗ . The operator ̂T is said to be a
partially normal extension of T .
Clearly,
subnormal =⇒ weakly subnormal =⇒ hyponormal. (5.34)
The converses of both implications in (5.34) are not true in general. Moreover, we
can easily see that the following statements are equivalent for T ∈ B(H):
(a) T is weakly subnormal;
(b) There is an extension ̂T of T such that ̂T ∗ ̂T f = ̂T ̂T ∗ f for all f ∈ H;
(c) There is an extension ̂T of T such that H ⊆ ker [ ̂T ∗ , ̂T ].
Weakly subnormal operators possess the following invariance properties:
(i) (Unitary equivalence) if T is weakly subnormal with a partially normal extension
( T 0 B A ) then for every unitary U, ( ) ( )
U ∗ T U U ∗ A
0 B (= U ∗ 0
0 I (
T A
0 B ) ( U 0 I 0 )) is a
partially normal extension of U ∗ T U, i.e., U ∗ T U is also weakly subnormal.
(ii) (Translation) if T ∈ L(H) is weakly subnormal then T − λ is also weakly subnormal
for every λ ∈ C: indeed if T has a partially normal extension ̂T then
̂T − λ := ̂T − λ satisfies the properties in Definition 5.3.21.
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