Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
Theorem 4.4.9. (Subnormal Extensions) Let W α be a subnormal weighted shift
with weights α : α 0 , α 1 , · · · and let µ be the corresponding Berger measure. Then
W (xn,··· ,x 1)α is subnormal if and only if
(i) 1
t
∈ L 1 (µ);
n
(ii) x j =
(iii) x n ≤
(
|| 1
t j−1 || L 1 (µ)
|| 1
t j || L 1 (µ)
(
|| 1
t n−1 || L 1 (µ)
|| 1
t n || L 1 (µ)
In particular, if we put
) 1
2
) 1
2
.
for 1 ≤ j ≤ n − 1;
S := {(x 1 , · · · , x n ) ∈ R n : W (xn,··· ,x 1)α
is subnormal}
then either S = ∅ or S is a line segment in R n .
Proof. Write W j := W (xn ,··· ,x 1 )α| ∨ {e n−j ,e n−j+1 ,··· } (1 ≤ j ≤ n) and hence W n =
W (xn ,··· ,x 1 )α. By the argument used to establish (3.2) we have that W 1 is subnormal
with associated measure ν 1 if and only if
(i) 1 t ∈ L1 (µ);
(ii) dν 1 = x2 1
t
dµ, or equivalently, x 2 1 =
( ∫ ||Wα || 2
0
) −1.
1
t dµ(t)
Inductively W n−1 is subnormal with associated measure ν n−1 if and only if
(i) W n−2 is subnormal;
(ii)
1
t n−1 ∈ L 1 (µ);
(iii) dν n−1 = x2 n−1
t
dν n−2 = · · · = x2 n−1···x2 1
t n−1 dµ, or equivalently, x 2 n−1 =
Therefore W n is subnormal if and only if
(i) W n−1 is subnormal;
(ii) 1
t
∈ L 1 (µ);
n
( ∫
(iii) x 2 ||Wα||
n ≤
2
0
) −1 (
1
∫
t dν ||Wα||
n−1 = 2
0
x 2 n−1···x2 1
t
dµ(t)
n
) −1
=
∫ ||W α|| 2
∫ ||W α|| 2 1
0
tn−2 dµ(t)
∫ ||Wα|| 2
1
0
tn−1 dµ(t).
1
0
tn−1 dµ(t)
∫ .
||W α|| 2 1
0
t n dµ(t)
Corollary 4.4.10. If W α is a subnormal weighted shift with associated measure µ,
there exists an n-step subnormal extension of W α if and only if
1
t n ∈ L 1 (µ).
Corollary 4.4.11. A recursively generated subnormal shift with φ 0 ≠ 0 admits an
n-step subnormal extension for every n ≥ 1.
Proof. The assumption about φ 0 implies that the zeros of g(t) are positive, so that
1
s 0 > 0. Thus for every n ≥ 1,
t
is integrable with respect to the corresponding
n
Berger measure µ = ρ 0 δ s0 + · · · + ρ r−1 δ sr−1 . By Corollary 4.4.10, there exists an
n-step subnormal extension.
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