Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
4.2 k-Hyponormality
Given an n-tuple T = (T 1 , . . . , T n ) of operators acting on H, we let
⎡
⎤
[T1 ∗ , T 1 ] [T2 ∗ , T 1 ] · · · [Tn, ∗ T 1 ]
[T ∗ [T1 ∗ , T 2 ] [T2 ∗ , T 2 ] · · · [Tn, ∗ T 2 ]
, T] ≡ ⎢
⎣
.
.
.
⎥
. ⎦ .
[T1 ∗ , T k ] [T2 ∗ , T k ] · · · [Tn, ∗ T n ]
By analogy with the case n = 1, we shall say that T is (jointly) hyponormal if
[T ∗ , T] ≥ 0.
An operator T ∈ B(H) is called k-hyponormal if (1, T, T 2 , · · · , T k ) is jointly hyponormal,
i.e.,
( ) k
M k (T ) ≡ [T ∗j , T i ]
i,j=1
⎡
[T ∗ , T ] [T ∗2 , T ] · · · [T ∗k ⎤
, T ]
[T ∗ , T 2 ] [T ∗2 , T 2 ] · · · [T ∗k , T 2 ]
= ⎢
⎥
⎣
.
.
.
. ⎦ ≥ 0
[T ∗ , T k ] [T ∗2 , T k ] · · · [T ∗k , T k ]
An application of Choleski algorithm for operator matrices shows that M k (T ) ≥ 0 is
equivalent to the positivity of the following matrix
⎡
1 T ∗ · · · T ∗k ⎤
T T ∗ T · · · T ∗k T
⎢
⎥
⎣
.
.
.
. ⎦ .
T k T ∗ T k · · · T ∗k T k
The Bram-Halmos criterion can be then rephrased as saying that
T is subnormal ⇐⇒ T is k-hyponormal for every k ≥ 1.
Recall ([Ath],[CMX],[CoS]) that T is called weakly k-hyponormal if
⎧
⎫
⎨ k∑
⎬
LS(T, T 2 , · · · , T k ) := α
⎩ j T j : α = (α 1 , · · · , α k ) ∈ C k ⎭
j=1
consists entirely of hyponormal operators, or equivalently, M k (T ) is weakly positive,
i.e.,
⎡ ⎤ ⎡ ⎤
⟨ λ 1 x λ 1 x ⟩
⎢
M k (T ) ⎣
⎥ ⎢
. ⎦ , ⎣
⎥
. ⎦ ≥ 0 ∀ λ 1 , · · · , λ k ∈ C.
λ k x λ k x
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