Woo Young Lee Lecture Notes on Operator Theory

Chapter 4
Weighted Shifts
4.1 Berger’s theorem
Recall that given a bounded sequence of positive numbers α : α 0 , α 1 , α 2 , · · · (called
weights), the (unilateral) weighted shift W α associated with α is the operator l 2 (Z + )
defined by
W α e n = α n e n+1 (n ≥ 0),
where {e n } ∞ n=0 is the canonical orthonormal basis for l 2 . It is straightforward to check
that
W α is compact ⇐⇒ α n → 0.
Indeed, W α = UD, where U is the unilateral shift and D is the diagonal operator
whose diagonal entries are α n .
We observe:
Proposition 4.1.1. If T ≡ W α is a weighted shift and ω ∈ ∂D then T ∼ = ωT .
Proof. If V e n := ω n e n for all n then V T V ∗ = ωT .
As a consequence of Proposition 4.1.1, we can see that the spectrum of a weighted
shift must be a circular symmetry:
Indeed we have:
σ(W α ) = σ(ωW α ) = ωσ(W α ).
Theorem 4.1.2. If T ≡ W α is a weighted shift with weight sequence α ≡ {α n } ∞ n=0
such that α n → α + then
(i) σ p (T ) = ∅;
(ii) σ(T ) = {λ : |λ| ≤ α + };
(iii) σ e (T ) = {λ : |λ| = α + };
(iv) |λ| < α + ⇒ ind (T − λ) = −1.
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