Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
We remember the following question (Due to P. Halmos):
Whether every polynomially hyponormal operator is subnormal
In 1993, R. Curto and M. Putinar [CP2] have answered it negatively:
There exits a polynomially hyponormal operator which is not 2-hyponormal.
In 1989, S. M. McCullough and V. Paulsen [McCP] proved the following: Every
polynomially hyponormal operator is subnormal if and only if every polynomially hyponormal
weighted shift is subnormal.
However we did not find a concrete example of such a weighted shift:
Problem 4.6. Find a weighted shift which is polynomially hyponormal but not subnormal.
Problem 4.7. Does there exists a polynomially hyponormal weighted shift which is
not 2-hyponormal
Let B 1 be the weighted shift whose weight are given by
√ x,
√
2
3 , √
5
4 , √
4
5 , · · ·
Let B 2 be the weighted shift whose weight are given by
√
1
2 , √ x,
√
3
4 , √
4
5 , · · ·
A straightforward calculation shows that
We conjecture that
B 1 subnormal ⇐⇒ 0 < x ≤ 1 2 ;
B 1 2-hyponormal ⇐⇒ 0 < x ≤ 9 16 ;
B 1 quadratically hyponormal ⇐⇒ 0 < x ≤ 2 3 ;
B 2 subnormal ⇐⇒ x = 2 3 ;
[
B 2 2-hyponormal ⇐⇒ x ∈
63 − √ 129
,
80
]
24
.
35
9
16 < sup{x : B 1 is polynomially hyponormal}
24
35 < sup{x : B 2 is polynomially hyponormal}
154