Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
Problem 4.8. Is the above converse true
We here suggest related problems:
Problem 4.9.
(a) Does there exists a Toeplitz operator which is polynomially hyponormal but not
subnormal
(b) Classify the polynomially hyponormal operators with finite rank self commutators.
(c) Is there an analogue of Berger’s theorem for polynomially hyponormal weighted
shift
An operator T ∈ B(H) is called M-hyponormal if
∃ M > 0 such that ||(T − λ) ∗ x|| ≤ M ||(T −λ)x|| for any λ ∈ C and for any x ∈ H.
If M ≤ 1 then M-hyponormality ⇒ hyponormality. It was shown [HLL] that it
T ≡ W α is a weighted shift with weight sequence α then
α is eventually increasing =⇒ T is hyponormal.
We wonder if the converse is also true.
Problem 4.10. (M-hyponormality of weighted shifts) Does it follow that
W α is M-hyponormal =⇒ α is eventually increasing
Problem 4.11 (Perturbations of weighted shifts) Let α be a strictly increasing weighted
sequence.
(a) If W α is k-hyponormal, dose it follow that W α is weakly k-hyponormal under
small perturbations of the weighted shifts
(b) Does it follow that the polynomiality of the weighted shifts is stable under small
perturbations of the weighted sequence
It was shown [CuL5] that the answer to Problem 4.10 (a) is affirmative if k = 2.
155