Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
For example, given α 0 < α 1 < α 2 , W (α0,α ̂ is the recursive weighted shift whose
1,α 2)
weights are calculated according to the recursive relation
αn+1 2 1
= φ 1 + φ 0
αn
2 ,
where φ 0 = − α2 0 α2 1 (α2 2 −α2 1 ) and φ
α 2 1 = − α2 1 (α2 2 −α2 0 ) . In this case, W
1 −α2 0
α 2 1 −α2 ̂
0
(α0
is subnormal
with 2-atomic Berger measure. Write W x(α0 ̂ for the weighted shift whose
,α 1 ,α 2 )
,α 1 ,α 2 )
weight sequence consists of the initial weight x followed by the weight sequence of
W
.
(α0,α ̂ 1,α 2)
By the Density Theorem ([CuF2, Theorem 4.2 and Corollary 4.3]), we know that
if W α is a subnormal weighted shift with weights α = {α n } and ϵ > 0, then there
exists a nonzero compact operator K with ||K|| < ϵ such that W α +K is a recursively
generated subnormal weighted shift; in fact W α + K = W for some m ≥ 1, where
̂α(m)
α (m) : α 0 , · · · , α m . The following result shows that K cannot generally be taken to
be finite rank.
Theorem 4.4.1. (Finite Rank Perturbations of Subnormal Shifts) If W α is a subnormal
weighted shift then there exists no nonzero finite rank operator F (≠ cP {e0 }) such
that W α + F is a subnormal weighted shift. Concretely, suppose W α is a subnormal
weighted shift with weight sequence α = {α n } ∞ n=0 and assume α ′ = {α n} ′ is a nonzero
perturbation of α in a finite number of weights except the initial weight; then W α ′ is
not subnormal.
We next consider the selfcommutator [(W α + s W 2 α) ∗ , W α + s W 2 α]. Let W α be a
hyponormal weighted shift. For s ∈ C, we write
D(s) := [(W α + s W 2 α) ∗ , W α + s W 2 α]
and we let
⎡
⎤
q 0 ¯r 0 0 . . . 0 0
r 0 q 1 ¯r 1 . . . 0 0
D n (s) := P n [(W α +s Wα) 2 ∗ , W α +s Wα]P 2 0 r 1 q 2 . . . 0 0
n =
⎢ .
.
.
. .. . . , (4.7)
⎥
⎣ 0 0 0 . . . q n−1 ¯r n−1
⎦
0 0 0 . . . r n−1 q n
where P n is the orthogonal projection onto the subspace generated by {e 0 , · · · , e n },
⎧
q n := u n + |s| 2 v n
⎪⎨ r n := s √ w n
u n := αn 2 − αn−1
2 (4.8)
v n := αnα 2 n+1 2 − αn−1α 2 n−2
2 ⎪⎩
w n := αn(α 2 n+1 2 − αn−1) 2 2 ,
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