Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
4.7 Comments and Problems
Problem 4.1. Let T x be a weighted shift with weights α ≡ {α n } given by
α : x,
√
2
3 , √
3
4 , √
4
5 , · · · .
Describe the set {x : T x is cubically hyponormal}. More generally, describe {x :
T x is weakly k-hyponormal}.
Problem 4.2. Let T be the weighted shift with weights α ≡ {α n } given by
If T is cubically hyponormal, is α flat
α 0 = α 1 ≤ α 2 ≤ α 3 ≤ · · · .
Problem 4.3. (Minimality of Weights Problem) If α : α 0 , α 1 , · · · , α 2k admits a
subnormal completion and if α ⊆ ω with W ω subnormal, does it follow that
α n ≤ ω n for all n ≥ 0
A combination of Theorem 4.6.2 (a) and (b) show that α n ≤ ω n for 0 ≤ n ≤ 2k + 1
and also for large n.
Problem 4.4. Given α : α 0 = α 1 < α 2 < · · · < α m , find necessary and sufficient
conditions for the existence of a quadratically hyponormal completion ω of α.
In [CuF2] it was shown that
∃ 1 < b < c such that W 1(1,
√
b,
√ c) ∧
is quadratically hyponormal. In fact, it was shown that if we write
H 2 := {(b, c) : W 1(1,
√
b,
√ c) ∧
is quadratically hyponormal}
then
where
H 2 := {(b, c) : b(bc − 1) + b(b − 1)(c − 1)K − (b − 1) 2 K 2 ≥ 0},
K =
b(c − 1)
(b(c 2 − 1) + √ )
b 2 (c − 1) 2 − 4b(b − 1)(c − b)
2(b − 1) 2 .
(c − b)
Problem 4.5. Does there exists 1 < b < c such that W 1,(1,
√
b,
√ c) ∧ is cubically
hyponormal More generally, describe the set
{(b, c) : W 1,(1,
√
b,
√ c) ∧
is cubically hyponormal}.
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