Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 3.
HYPONORMAL AND SUBNORMAL THEORY
We should prove that P k → 1(SOT). Since {P k } is increasing, L = cl ∪ k ranP k is a
closed linear space. To show that P k → 1(SOT) it suffices to show that L = H. To
do this, it suffices to show that f(T )L ⊆ L for all f ∈ Rat(σ(T )). Since {λ j } is dense
in σ(T ) c , it is only necessary to show that f(T )L ⊆ L when f is a rational function
with poles in {λ j }. Hence we must show that
T L ⊆ L and (T − λ j ) −1 L ⊆ L.
From the definition of L we see that these two conditions are equivalent, respectively,
to show that for all β ≥ 1:
)
T
(T j (T − λ 1 ) −1 · · · (T − λ k ) −1 g i ∈ L for 0 ≤ j ≤ 2k ; (3.1)
(T − λ m ) −1 (
T j (T − λ 1 ) −1 · · · (T − λ k ) −1 g i
)
∈ L for 0 ≤ j ≤ 2k and all m. (3.2)
To prove (3.1) we need only consider the case where j = 2k. Now
T 2k+1 (T − λ 1 ) −1 · · · (T − λ 2k ) −1 g i ∈ ranP 2k
and A = (T − λ k+1 ) · · · (T − λ 2k ) is a polynomial in T of degree 2k − k. Hence
T 2k+1 (T −λ 1 ) −1 · · · (T −λ k ) −1 g i = AT 2k+1 (T −λ 1 ) −1 · · · (T −λ 2k ) −1 g i ∈ ranP 2k ⊆ L,
which proves (3.1).
Since (3.1) implies that L is an invariant subspace for T , to show (3.2) it suffices
to show that
(
)
(T − λ m ) −1 (T − λ 1 ) −1 · · · (T − λ k ) −1 g i ∈ L for all m.
Since λ m is repeated infinitely often, we may assume m ≥ k + 2. If B = (T −
λ k+1 ) · · · (T − λ m−1 ), then B is a polynomial in T of degree m + k − 1. Hence
(
)
(T −λ m ) −1 (T −λ 1 ) −1 · · · (T −λ k ) −1 g i = B(T −λ 1 ) −1 · · · (T −λ m ) −1 g i ∈ ranP m ⊆ L,
which proves (3.2).
Lemma 3.2.6. If T ∈ B(H) is an m-multicyclic hyponormal operator then
tr [T ∗ , T ] ≤ m∥T ∥ 2 .
Proof. By Lemma 3.2.5, there exists an increasing 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. Note that
∥P ⊥ k T P k ∥ 2 2 ≤ m∥P ⊥ k T P k ∥ 2 .
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