Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
and, for notational convenience, α −2 = α −1 = 0. Clearly, W α is quadratically hyponormal
if and only if D n (s) ≥ 0 for all s ∈ C and all n ≥ 0. Let d n (·) := det (D n (·)).
Then d n satisfies the following 2–step recursive formula:
d 0 = q 0 , d 1 = q 0 q 1 − |r 0 | 2 , d n+2 = q n+2 d n+1 − |r n+1 | 2 d n .
If we let t := |s| 2 , we observe that d n is a polynomial in t of degree n + 1, and if
we write d n ≡ ∑ n+1
i=0 c(n, i)ti , then the coefficients c(n, i) satisfy a double-indexed
recursive formula, namely
c(n + 2, i) = u n+2 c(n + 1, i) + v n+2 c(n + 1, i − 1) − w n+1 c(n, i − 1),
c(n, 0) = u 0 · · · u n , c(n, n + 1) = v 0 · · · v n , c(1, 1) = u 1 v 0 + v 1 u 0 − w 0
(4.9)
(n ≥ 0, i ≥ 1). We say that W α is positively quadratically hyponormal if c(n, i) ≥ 0
for every n ≥ 0, 0 ≤ i ≤ n + 1. Evidently, positively quadratically hyponormal =⇒
quadratically hyponormal. The converse, however, is not true in general.
The following theorem establishes a useful relation between 2-hyponormality and
positive quadratic hyponormality.
Theorem 4.4.2. Let α ≡ {α n } ∞ n=0 be a weight sequence and assume that W α is 2-
hyponormal. Then W α is positively quadratically hyponormal. More precisely, if W α
is 2-hyponormal then
c(n, i) ≥ v 0 · · · v i−1 u i · · · u n (n ≥ 0, 0 ≤ i ≤ n + 1). (4.10)
In particular, if α is strictly increasing and W α is 2-hyponormal then the Maclaurin
coefficients of d n (t) are positive for all n ≥ 0.
If W α is a weighted shift with weight sequence α = {α n } ∞ n=0, then the moments
of W α are usually defined by β 0 := 1, β n+1 := α n β n (n ≥ 0); however, we prefer to
reserve this term for the sequence γ n := βn 2 (n ≥ 0). A criterion for k-hyponormality
can be given in terms of these moments ([Cu2, Theorem 4]): if we build a (k + 1) ×
(k + 1) Hankel matrix A(n; k) by
⎡
⎤
γ n γ n+1 . . . γ n+k
γ n+1 γ n+2 . . . γ n+k+1
A(n; k) := ⎢
⎣
.
.
⎥ (n ≥ 0),
. ⎦
γ n+k γ n+k+1 . . . γ n+2k
then
W α is k-hyponormal ⇐⇒ A(n; k) ≥ 0 (n ≥ 0). (4.11)
In particular, for α strictly increasing, W α is 2-hyponormal if and only if
⎡
⎤
γ n γ n+1 γ n+2
det ⎣γ n+1 γ n+2 γ n+3
⎦ ≥ 0 (n ≥ 0). (4.12)
γ n+2 γ n+3 γ n+4
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