Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
One might conjecture that if W α is a k-hyponormal weighted shift whose weight
sequence is strictly increasing then W α remains weakly k-hyponormal under a small
perturbation of the weight sequence. We will show below that this is true for k = 2
([]).
In [CuF3, Theorem 4.3], it was shown that the gap between 2-hyponormality and
quadratic hyponormality can be detected by unilateral shifts with a weight sequence
α : √ x, ( √ a, √ b, √ c) ∧ . In particular, there exists a maximum value H 2 ≡ H 2 (a, b, c)
of x that makes W √ x,( √ a, √ b, √ c)
2-hyponormal; H ∧ 2 is called the modulus of 2-
hyponormality (cf. citeCuF3). Any value of x > H 2 yields a non-2-hyponormal
weighted shift. However, if x − H 2 is small enough, W √ x,( √ a, √ b, √ c)
is still quadratically
hyponormal. The following theorem shows that, more generally, for finite rank
∧
perturbations of weighted shifts with strictly increasing weight sequences, there always
exists a gap between 2-hyponormality and quadratic hyponormality.
Theorem 4.4.3. (Finite Rank Perturbations of 2-hyponormal Shifts) Let α = {α n } ∞ n=0
be a strictly increasing weight sequence. If W α is 2-hyponormal then W α remains positively
quadratically hyponormal under a small nonzero finite rank perturbation of α.
We are ready for:
Proof of Theorem 4.4.1. It suffices to show that if T is a weighted shift whose restriction
to ∨ {e n , e n+1 , · · · } (n ≥ 2) is subnormal then there is at most one α n−1 for
which T is subnormal.
Let W := T | ∨ {e n−1 ,e n ,e n+1 ,··· } and S := T | ∨ {e n ,e n+1 ,··· }, where n ≥ 2. Then
W and S have weights α k (W ) := α k+n−1 and α k (S) := α k+n (k ≥ 0). Thus the
corresponding moments are related by the equation
γ k (S) = αn 2 · · · αn+k−1 2 = γ k+1(W )
αn−1
2 .
We now adapt the proof of [Cu2, Proposition 8]. Suppose S is subnormal with associated
Berger measure µ. Then γ k (S) = ∫ ||T || 2
t k dµ. Thus W is subnormal if and
0
only if there exists a probability measure ν on [0, ||T || 2 ] such that
1
α 2 n−1
∫ ||T ||
2
0
t k+1 dν(t) =
∫ ||T ||
2
0
t k dµ(t) for all k ≥ 0,
which readily implies that t dν = α 2 n−1 dµ. Thus W is subnormal if and only if the
formula
dν := λ · δ 0 + α2 n−1
dµ
t
defines a probability measure for some λ ≥ 0, where δ 0 is the point mass at the
origin. In particular 1 t
∈ L1 (µ) and µ({0}) = 0 whenever W is subnormal. If we
repeat the above argument for W and V := T | ∨ {e n−2,e n−1,··· }, then we should have
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