Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
we have
⎡
⎡
1 x 2 2 3 x2
det ⎣ x 2 2 3 x2 1 2 x2 ⎦ = x 2 det ⎣
⎤
2
3 x2 1 2 x2 2 5 x2
1 2
x
1 2 3
2 1
1
3
2 1 2
3 2 5
= x 2 ( 1
60x 2 − 4
135
⎤
⎦
)
≥ 0 =⇒ x ≤ 3 4 .
(c) See [Cu2]
Let W α be a weighted shift with weights α ≡ {α n } ∞ n=0. For s ∈ C, write
D(s) :=
[ (Wα
+ sWα
2 ) ∗
, Wα + sWα]
2
and let
⎡
⎤
q 0 γ 0 0 · · · 0 0
γ 0 q 1 γ 1 · · · 0 0
0 γ 1 q 2 · · · 0 0
D n (s) := P n D(s)P n =
.
⎢ . . . ..
. ,
.
⎥
⎣ 0 0 0 · · · q n−1 γ n−1
⎦
0 0 0 · · · γ n−1 q n
where P n := the orthogonal projection onto the subspace spanned by {e 0 , · · · , e n },
where
⎧
⎨
⎩
{
qn := u n + |s| 2 v n
γ n := s √ w n ,
u n := α 2 n − α 2 n−1
v n := α 2 nα 2 n+1 − α 2 n−1α 2 n−2
w n = α 2 n(α 2 n+1 − α 2 n−1) 2 ,
and, for notational convenience, α −2 = α −1 = 0.
Clearly,
W α is quadratically hyponormal ⇐⇒ D n (s) ≥ 0 for any s ∈ C, for any n ≥ 0.
Let d n (·) = detD n (·). Then d n satisfies the following 2-step recursive formula:
d 0 = q 0 , d 1 = q 0 q 1 − |γ 0 | 2 , d n+2 = q n+2 d n+1 − |γ n+1 | 2 d n .
If we let t := |s| 2 , we observe that d n is a polynomial in t of degree n + 1. If we write
n+1
∑
d n ≡ c(n, i)t i ,
i=0
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