Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
Proof. The assertion (i) is straightforward. For the other assertions, observe that if
α + = 0 then T is compact and quasinilpotent. If instead α + > 0 then T − α + U
(U :=the unilateral shift) is a weighted shift whose weight sequence converges to 0.
Hence T − α + U is a compact and hence
σ e (T ) = σ e (α + U) = α + σ e (U) = {λ : |λ| = α + }.
If |λ| < α + then T − λ is Fredholm and
index (T − λ) = index (α + U − λ) = −1.
In particular, {λ : |λ| ≤ α + } ⊂ σ(T ). By the assertion (i), we can conclude that
σ(T ) = {λ : |λ| ≤ α + }.
Theorem 4.1.3. If T ≡ W α is a weighted shift with weight sequence α ≡ {α n } ∞ n=0
then
⎡
⎤
α0
2 [T ∗ α1 2 − α 2 0
, T ] = ⎢
⎣
α2 2 − α1
2 ⎥
⎦
. ..
Proof. From a straightforward calculation.
The moments of W α are defined by
β 0 := 1, β n+1 = α 0 · · · α n ,
but we reserve this term for the sequence γ n := βn.
2
Theorem 4.1.4. (Berger’s theorem) Let T ≡ W α be a weighted shift with weight
sequence α ≡ {α n } and define the moment of T by
γ 0 := 1 and γ n := α 2 0α 2 1 · · · α 2 n−1 (n ≥ 1).
Then T is subnormal if and only if there exists a probability measure ν on [ 0, ∥T ∥ 2]
such that
∫
γ n = t n dν(t) (t ≥ 1). (4.1)
[0,∥T ∥ 2 ]
Proof. (⇒) Note that T is cyclic. So if T is subnormal then T ∼ = S µ , i.e., there is an
isomorphism U : L 2 (µ) −→ P 2 (µ) such that
Ue 0 = 1 and UT U −1 = S µ .
Observe T n e 0 = √ γ n e n for all n. Also U(T n e 0 ) = SµUe n 0 = Sµ1 n = z n . So
∫
∫
∫
|z| 2n dµ = |UT n e 0 | 2 dµ = |U( √ ∫
γ n e n )| 2 dµ = γ n |Ue n | 2 dµ = γ n ∥Ue n ∥ = γ n .
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