Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
Proof. [T ∗ , T] ≥ 0 ⇐⇒
Thus
[T ∗ , T ∗ ] ≥ 0 ⇐⇒
⟨ ( x
[T ∗ , T ∗ ]
ty
) ( x
,
ty
⟨[ [T
∗
1 , T 1 ] [T ∗ 2 , T 1 ]
[T ∗ 1 , T 2 ] [T ∗ 2 , T 2 ]] ( x
ty
)⟩
≥ 0 for any x, y ∈ H and t ∈ R.
) ( x
,
ty
)⟩
≥ 0
⇐⇒⟨[T ∗ 1 , T 1 ]x, x⟩ + t 2 ⟨[T ∗ 2 , T 2 ]y, y⟩ + 2tRe ⟨[T ∗ 2 , T 1 ]y, x⟩ ≥ 0 (†)
Conversely if (*) holds then
which implies (†) holds.
=⇒If T 1 and T 2 are hyponormal then
t 2 ⟨[T ∗ 2 , T 2 ]y, y⟩ + 2t ∣ ∣⟨[T ∗ 2 , T 1 ]y, x⟩ ∣ ∣ + ⟨[T ∗ 1 , T 1 ]x, x⟩ ≥ 0
=⇒D/4 ≡ |⟨[T ∗ 2 , T 1 ]y, x⟩| 2 − ⟨[T ∗ 1 , T 1 ]x, x⟩⟨[T ∗ 2 , T 2 ]y, y⟩ ≤ 0
=⇒ ∣ ∣⟨[T ∗ 2 , T 1 ]y, x⟩ ∣ ∣ 2 ≤ ⟨[T ∗ 1 , T 1 ]x, x⟩⟨[T ∗ 2 , T 2 ]y, y⟩ (∗)
Re ⟨[T ∗ 2 , T 1 ]y, x⟩ 2 ≤ ⟨[T ∗ 1 , T 1 ]x, x⟩⟨[T ∗ 2 , T 2 ]y, y⟩,
Corollary 4.2.3. Let T = (T 1 , T 2 ) be a pair of operators on H. Then T is hyponormal
if and only if T 1 and T 2 are hyponormal and
[T2 ∗ , T 1 ] = [T1 ∗ , T 1 ] 1 2 D[T
∗
2 , T 2 ] 1 2
for some contraction D.
Proof. This follows from a theorem of Smul’jan [Smu]:
[ ] A B
B ∗ ≥ 0 ⇐⇒ A ≥ 0, C ≥ 0 and B = √ AD √ C for some contraction D.
C
Corollary 4.2.4. Let T ≡ W α be a weighted shift with weight sequence α :
α 1 ≤ α 2 ≤ · · · . Then the following are equivalent:
α 0 ≤
(i) T is 2-hyponormal;
(
(ii) αn+1 2 α
2
n+2 − αn) 2 2 ( (
≤ α
2
n+1 − αn) 2 α
2
n+2 αn+3 2 − αnαn+1) 2 2 for any n ≥ 0;
(
(iii) αn 2 α
2
n+2 − αn+1) 2 2 ( (
≤ α
2
n+2 α
2
n+1 − αn) 2 α
2
n+3 − αn+2) 2 for any n ≥ 0.
Proof. By Corollary 4.2.3,
(T, T 2 ) hyponormal ⇐⇒ [T ∗2 , T ] = [T ∗ , T ] 1 2 E[T ∗2 , T 2 ] 1 2 for some contraction E.
111