Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
If ν is defined on [0, ||T || 2 ] by
ν(∆) := µ ( {z : |z| 2 ∈ ∆} ) ,
then ν is a probability measure and γ n = ∫ t n dν(t).
(⇐) If ν is the measure satisfying (4.1), define the measure µ by dµ(re iθ ) =
1
2π dθdµ(r). Then we can see that T ∼ = S µ .
Example 4.1.5. (a) The Bergman shift B α is the weighted shift with weight sequence
α ≡ {α n } given by
√
n + 1
α n = (n ≥ 0).
n + 2
Then B α is subnormal: indeed,
γ n := α 2 0α 2 1 · · · α 2 n−1 = 1 2 · 2
3 · · · n
n + 1 = 1
n + 1
and if we define µ(t) = t, i.e., dµ = dt then
∫ 1
0
t n dµ(t) = 1
n + 1 = γ n.
(b) If α n : β, 1, 1, 1, · · · then W α is subnormal: indeed γ n = β 2 and if we define
dµ = β 2 δ 1 + (1 − β 2 )δ 0 then ∫ 1
0 tn dµ = β 2 = γ n .
Remark. Recall that the Bergman space A(D) for D is defined by
{
∫
}
A(D) := f : D → C : f is analytic with |f| 2 dµ < ∞ .
Then the orthonormal basis for A(D) is given by {e n ≡ √ n + 1 z n : n = 0, 1, 2, · · · }
with dµ = 1 π
dA. The Bergman operator T : A(D) → A(D) is defined by
T f = zf.
In this case the matrix (α ij ) of the Bergman operator T with respect to the basis
{e n ≡ √ n + 1 z n : n = 0, 1, 2, · · · } is given by
α ij = ⟨T e j , e i ⟩
= ⟨T √ j + 1 z j , √ i + 1 z i ⟩
= ⟨ √ j + 1 z j+1 , √ i + 1 z i ⟩
= √ ∫
(j + 1)(i + 1) z j+1 z i dµ
D
∫ 2π ∫ 1
= √ (j + 1)(i + 1) 1 π 0
{√
j+1
=
j+2
(i = j + 1)
0 (i ≠ j + 1) :
107
0
D
r j+1+i e i(j+1−i)θ · rdr dθ