Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
This completes the proof.
□
To prove Theorem 4.4.3 we need:
Lemma 4.4.7. ([CuF3, Lemma 2.3]) Let α ≡ {α n } ∞ n=0 be a strictly increasing weight
sequence. If W α is 2-hyponormal then the sequence of quotients
Θ n := u n+1
u n+2
(n ≥ 0)
is bounded away from 0 and from ∞. More precisely,
1 ≤ Θ n ≤ u 1
u 2
( ||Wα || 2
α 0 α 1
) 2
for sufficiently large n.
In particular, {u n } ∞ n=0 is eventually decreasing.
We are ready for:
Proof of Theorem 4.4.3. By Theorem 4.4.2, W α is strictly positively quadratically
hyponormal, in the sense that all coefficients of d n (t) are positive for all n ≥ 0. Note
that finite rank perturbations of α affect a finite number of values of u n , v n and w n .
More concretely, if α ′ is a perturbation of α in the weights {α 0 , · · · , α N }, then u n , v n ,
w n and p n are invariant under α ′ for n ≥ N + 3. In particular, p n ≥ 0 for n ≥ N + 3.
Claim 1. For n ≥ 3, 0 ≤ i ≤ n + 1,
c(n, i) =u n c(n − 1, i) + p n−1 c(n − 2, i − 1) +
where
+ v n · · · v 3 ρ i−n+1 ,
(cf. [CuF3, Proof of Theorem 4.3]).
n∑
k=4
p k−2
⎛
⎝
n∏
j=k
⎧
0 (i < n − 1)
⎪⎨
u 0 p 1 (i = n − 1)
ρ i−n+1 =
v 0 p 1 + v 2 p 0 (i = n)
⎪⎩
v 0 v 1 v 2 (i = n + 1)
Proof of Claim 1. We use induction. For n = 3, 0 ≤ i ≤ 4,
v j
⎞
⎠ c(k − 3, i − n + k − 2)
(4.19)
c(3, i) = u 3 c(2, i) + v 3 c(2, i − 1) − w 2 c(1, i − 1)
)
= u 3 c(2, i) + v 3
(u 2 c(1, i − 1) + v 2 c(1, i − 2) − w 1 c(0, i − 2) − w 2 c(1, i − 1)
)
= u 3 c(2, i) + p 2 c(1, i − 1) + v 3
(v 2 c(1, i − 2) − w 1 c(0, i − 2)
= u 3 c(2, i) + p 2 c(1, i − 1) + v 3 ρ i−2 ,
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