Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 3.
HYPONORMAL AND SUBNORMAL THEORY
so
tr [T ∗ m
, T ] = 1,
π Area(σ(T )) = 1 π · π = 1.
To prove Theorem 3.2.3 we need auxiliary lemmas. Recall the Hilbert-Schmidt norm
of X:
[∑ ⟨
∥X∥ 2 ≡ |X| 2 ⟩ ] 1 2
e n , e n
[∑
1
= ⟨X ∗ 2
Xe n , e n ⟩]
= [tr (X ∗ X)] 1 2
.
Lemma 3.2.4. If T ∈ B(H) and P is a finite rank projection then
Proof. Write
Since P =
[ ] 1 0
,
0 0
tr ( P [T ∗ , T ]P ) ≤ ∥P ⊥ T P ∥ 2 2.
[ ] A B
T = :
C P
[ ] [ ]
ranP ranP
ranP ⊥ →
ranP ⊥ .
P [T ∗ , T ]P = [A ∗ , A] + C ∗ C − BB ∗ .
So by the above remark, tr (P [T ∗ , T ]P ) = tr[A ∗ , A] + ∥C∥ 2 2 − ∥B∥ 2 2. But since A is a
finite-dimensional operator,
tr[A ∗ , A] = 0.
Hence tr (P [T ∗ , T ]P ) ≤ ∥C∥ 2 2 = ||P ⊥ T P || 2 2.
Lemma 3.2.5. If T ∈ B(H) is an m-multicyclic operator then there exists a sequence
{P k } of finite rank projections such that P k ↑ 1(SOT) and
rank ( P ⊥ k T P k
)
≤ m for all k ≥ 1.
Proof. Let g 1 , · · · , g m be the generating vectors for T and let {λ j } be a countable
dense subset of C \ σ(T ); for convenience, arrange {λ j } so that each point is repeated
infinitely often. Let P k be the projection of H onto
∨ {
T j (T − λ 1 ) −1 · · · (T − λ k ) −1 g i : 0 ≤ j ≤ 2k, 1 ≤ i ≤ m } .
Thus P k is finite rank, P k ≤ P k+1 , and
rank [P ⊥ k T P k ] ≤ m for all k ≥ 1.
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