Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 3.
HYPONORMAL AND SUBNORMAL THEORY
Since {P k } is an increasing sequence,
tr[T ∗ , T ] = lim k tr ( P k [T ∗ )
, T ]P k .
By Lemma 3.2.4 we get
)
tr [T ∗ , T ] ≤ limsup∥Pk ⊥ T P k ∥ 2 2 ≤ limsup
(m∥Pk ⊥ T P k ∥ 2 ≤ m∥T ∥ 2 .
We are ready for:
Proof of the Berger-Shaw Theorem. Let R = ∥T ∥ and put D = B(0; R). If ε > 0, let
D 1 , · · · , D n be pairwise disjoint closed disks contained in D \ σ(T ) such that
Area (D) < Area σ(T ) + ∑ j
Area (D j ) + ε.
If D j = B(a j ; r j ), this inequality says
πR 2 − π ∑ j
r 2 j < Area σ(T ) + ε.
If S is the unilateral shift of multiplicity 1, let S j = (a j + r j S) (m) . Now that each S j
is m-multicyclic. Thus
⎡
⎤
T
S 1 0
A = ⎢
⎣
.
0 ..
⎥
⎦
S n
is an m-multicyclic hyponormal operator since the spectra of the operator summands
are pairwise disjoint. Also ∥A∥ = R. By Lemma 3.2.6, tr [A ∗ , A] ≤ mR 2 . But
Therefore
tr [A ∗ , A] = tr [T ∗ , T ] +
⎛
n∑
n∑
tr [Sj ∗ , S j ] = tr [T ∗ , T ] + m rj 2 .
j=1
π tr [T ∗ , T ] ≤ m ⎝πR 2 − π
n∑
j=1
Since ε was arbitrary, the proof is complete.
r 2 j
⎞
j=1
⎠ ≤ m ( Area σ(T ) + ε ) .
Theorem 3.2.7. (Putnam’s inequality) If S ∈ B(H) is a hyponormal operator then
∥[S ∗ , S]∥ ≤ 1 Area (σ(S)).
π
84