Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
Therefore if W α is 2-hyponormal then by Lemma 4.4.7, the sequences
{ α
2
n+1 − α 2 n
α 2 n − α 2 n−1
} ∞
n=2
and
{ α
2
n−1 − α 2 n−2
α 2 n − α 2 n−1
} ∞
n=2
are both bounded, so that {k n } ∞ n=2 is bounded. This proves Claim 2.
Write k := sup n k n . Without loss of generality we assume k < 1 (this is possible
from the observation that cα induces {c 2 k n }). Choose a sufficiently small perturbation
α ′ of α such that if we let
h :=
sup
0≤l≤N+2;0≤m≤1
N+4
∑
∣
k=4
p ′ k−2
⎛
∏
⎝
N+3
v j
′
j=k
⎞
⎠ c ′ (k − 3, l) + v N+3 ′ · · · v 3 ′ ρ ′ m
∣
(4.21)
then
c ′ (N + 3, i) − 1 h > 0 (0 ≤ i ≤ N + 3) (4.22)
1 − k
(this is always possible because by Theorem 4.4.2, we can choose a sufficiently small
|p ′ i | such that
c ′ (N + 3, i) > v 0 · · · v i−1 u i · · · u N+3 − ϵ and |h| < (1 − k) ( v 0 · · · v i−1 u i · · · u N+3 − ϵ )
for any small ϵ > 0).
Claim 3. For j ≥ 4 and 0 ≤ i ≤ N + j,
)
∑j−3
c ′ (N + j, i) ≥ u N+j · · · u N+4
(c ′ (N + 3, i) − k n h . (4.23)
Proof of Claim 3. We use induction. If j = 4 then by Claim 1 and (4.21),
c ′ (N + 4, i) = u ′ N+4c ′ (N + 3, i) + p ′ N+3c ′ (N + 2, i − 1)
⎛ ⎞
N+4
∑
N+3
∏
+ v N+4
′ p ′ ⎝
k−2 v j
′ ⎠ c ′ (k − 3, i − N + k − 6) + v N+4 ′ · · · v 3ρ ′ ′ i−(N+3)
k=4
j=k
n=1
≥ u ′ N+4c ′ (N + 3, i) + p ′ N+3c ′ (N + 2, i − 1) − v ′ N+4h
≥ u N+4
(
c ′ (N + 3, i) − k N+4 h )
≥ u N+4
(
c ′ (N + 3, i) − k h )
because u ′ N+4 = u N+4, v ′ N+4 = v N+4 and p ′ N+3 = p N+3 ≥ 0. Now suppose (4.23)
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